arXiv:1706.04387v1 [] 14 Jun 2017

TOPOLOGICAL FINITENESS PROPERTIES OF MONOIDS
PART 1: FOUNDATIONS
ROBERT D. GRAY AND BENJAMIN STEINBERG
Abstract. We initiate the study of higher dimensional topological finiteness properties of
monoids. This is done by developing the theory of monoids acting on CW complexes. For this
we establish the foundations of M -equivariant homotopy theory where M is a discrete monoid.
For projective M -CW complexes we prove several fundamental results such as the homotopy
extension and lifting property, which we use to prove the M -equivariant Whitehead theorems.
We define a left equivariant classifying space as a contractible projective M -CW complex. We
prove that such a space is unique up to M -homotopy equivalence and give a canonical model for
such a space via the nerve of the right Cayley graph category of the monoid. The topological
finiteness conditions left-Fn and left geometric dimension are then defined for monoids in terms
of existence of a left equivariant classifying space satisfying appropriate finiteness properties.
We also introduce the bilateral notion of M -equivariant classifying space, proving uniqueness
and giving a canonical model via the nerve of the two-sided Cayley graph category, and we define
the associated finiteness properties bi-Fn and geometric dimension. We explore the connections
between all of the these topological finiteness properties and several well-studied homological
finiteness properties of monoids which are important in the theory of string rewriting systems,
including FPn , cohomological dimension, and Hochschild cohomological dimension. We also
develop the corresponding theory of M -equivariant collapsing schemes (that is, M -equivariant
discrete Morse theory), and among other things apply it to give topological proofs of results of
Anick, Squier and Kobayashi that monoids which admit presentations by complete rewriting
systems are left- right- and bi-FP∞ .

1. Introduction
The study of the higher dimensional finiteness properties of groups was initiated fifty years
ago by C. T. C. Wall [Wal65] and Serre [Ser71]. An Eilenberg–MacLane complex K(G, 1) for a
discrete group G, also called a classifying space, is an aspherical CW complex with fundamental
group G. Such a space can always be constructed for any group G (e.g. via the bar construction) and it is unique up to homotopy equivalence. While useful for theoretical purposes, this
canonical K(G, 1)-complex is very big and is often not useful for practical purposes, specifically
if one wants to compute the homology of the group. It is therefore natural to seek a ‘small’
K(G, 1) for a given group by imposing various finiteness conditions on the space. Two of the
most natural and well-studied such conditions are the topological finiteness property Fn and
the geometric dimension gd(G) of the group.
Property Fn was introduced by C. T. C. Wall in [Wal65]. A group G is said to be of type
Fn if it has an Eilenberg–MacLane complex K(G, 1) with finite n-skeleton. It is easily verified
that a group is finitely generated if and only if it is of type F1 and is finitely presented if
Date: June 15, 2017.
2010 Mathematics Subject Classification. 20M50, 20M05, 20J05, 57M07, 20F10, 20F65.
Key words and phrases. CW complex, monoid, equivariant homotopy theory, homological finitenss property
FPn , cohomological dimension, Hoschild cohomology, rewriting system, collapsing scheme, discrete Morse theory.
This work was supported by the EPSRC grant EP/N033353/1 ‘Special inverse monoids: subgroups, structure,
geometry, rewriting systems and the word problem’. The second author was supiported in part by United StatesIsrael Binational Science Foundation #2012080 and NSA MSP #H98230-16-1-0047.
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ROBERT D. GRAY AND BENJAMIN STEINBERG

and only if it is of type F2 . Thus property Fn generalises the two fundamental finiteness
properties of being finitely generated, or finitely presented, to higher dimensions. The geometric
dimension of G, denoted gd(G), is the smallest non-negative integer n such that there exists
an n-dimensional K(G, 1) complex. If no such n exists, then we set gd(G) = ∞. For more
general background on higher dimensional finiteness properties of groups we refer the reader to
the books [Bro94, Chapter 8], [Geo08, Chapters 6-9], or the survey article [Bro10].
Each of these topological finiteness properties has a natural counterpart in homological algebra given in terms of the existence of projective resolutions of ZG-modules. The analogue
of Fn in this context is the homological finiteness property FPn , while geometric dimension
corresponds to the cohomological dimension of the group. Recall that a group G is said to be
of type FPn (for a positive integer n) if there is a projective resolution P of Z over ZG such
that Pi is finitely generated for i ≤ n. We say that G is of type FP∞ if there is a projective
resolution P of Z over ZG with Pi finitely generated for all i. The property FPn was introduced
for groups by Bieri in [Bie76] and since then has received a great deal of attention in the literature; see [BB97,BH01,Bra99,BW07,LS06]. For groups, Fn and FPn are equivalent for n = 0, 1,
while important results of Bestvina and Brady [BB97] show that FP2 is definitely weaker than
F2 . For higher n there are no further differences, in that a group G is of type Fn (2 ≤ n ≤ ∞)
if and only if it is finitely presented and of type FPn . One natural line of investigation has
been the study of the closure properties of FPn . Examples include results about the behaviour
of FPn under taking: finite index subgroups or extensions, direct (and semidirect) products,
wreath products, HNN extensions, amalgamated free products, and quasi-isometry invariance;
see for example [Alo94, BBG98, Bie76]. In [BHMS02] it is shown that if G is a subgroup of a
direct product of n surface groups, then if G is of type FPn then G has a subgroup of finite
index which is the direct product of at most n finitely generated surface groups. Thompson’s
groups, and several interesting generalisations of these groups, have all be shown to be of type
FP∞ ; see for example [Bro87, FMWZ13, Ste92, GS06].
The cohomological dimension of a group G, denoted cd(G), is the smallest non-negative
integer n such that there exists a projective resolution P = (Pi )i≥0 of Z over ZG of length ≤ n,
i.e., satisfying Pi = 0 for i > n. (Or, if no such n exists, then we set cd(G) = ∞.) The geometric
dimension of a group provides an upper bound for the cohomological dimension. It is easily
seen that gd(G) = cd(G) = 0 if and only if G is trivial. It follows from important results of
Stallings [Sta68] and Swan [Swa69] that gd(G) = cd(G) = 1 if and only if G is non-trivial free
group. Eilenberg and Ganea [EG57] proved that for n ≥ 3 the cohomological and the geometric
dimension of a group are the same. The famous Eilenberg–Ganea problem asks whether this
also holds in dimension two.
Working in the more general context of monoids, and projective resolutions of left ZMmodules, gives the notion of left-FPn , and left cohomological dimension, of a monoid M . There
is an obvious dual notion of monoids of type right-FPn , working with right ZM -modules.
Working instead with bimodules resolutions of the (ZM , ZM )-bimodule ZM one obtains the
notion bi-FPn introduced and studied in [KO01]. Property bi-FPn is of interest from the point of
view of Hochschild cohomology, which is the standard notion of cohomology for rings; [Hoc45],
[Wei94, Chapter 9], or [Mit72]. For monoids all these notions of FPn are known to be different,
while for groups they are all equivalent; see [Coh92, Pri06]. Similarly there is a dual notion of
the right cohomological dimension of a monoid which again is in general not equal to the left
cohomological dimension; see [GP98]. The two-sided notion is the Hochschild cohomological
dimension [Mit72].

TOPOLOGICAL FINITENESS PROPERTIES

3

In monoid and semigroup theory the property FPn arises naturally in the study of string
rewriting systems (i.e. semigroup presentations). The history of rewriting systems in monoids and groups is long and distinguished, and has roots in fundamental work of Dehn and
Thue. A central topic in this area is the study of complete rewriting systems and in methods for computing normal forms. A finite complete rewriting system is a finite presentation for a monoid of a particular form (both confluent and Noetherian) which in particular
gives a solution of the word problem for the monoid; see [BO93]. It is therefore of considerable interest to develop an understanding of which monoids are presentable by such rewriting systems. Many important classes of groups are known to be presentable by finite complete rewriting systems, including surface groups, Coxeter groups, and many closed threemanifold groups. Rewriting systems continue to receive a lot of attention in the literature;
see [CS09a,CS09b,Cho06,DDM09,GS08,HS99,PO05]. The connection between complete rewriting systems and homological finiteness properties is given by a result of Anick [Ani86] (see
also [Bro92]) which shows that a monoid that admits such a presentation must be of type
left- and right-FP∞ ; the special case of FP3 was also handled by Squier [Squ87]. More generally Kobayashi [Kob05] proved that any such monoid is of type bi-FP∞ . A range of other
interesting homotopical and homological finiteness properties have been studied in relation to
monoids defined by compete rewriitng systems including finite homological type, finite derivation type, and higher dimensional analogues FDTn ; see [SOK94, PO04, PO05, GM12]. More
background on the importance the property FPn (and other related finiteness conditions) in
semigroup theory, and the connections with the theory of string rewriting systems may be found
in the survey articles [Coh97, OK97]. Results on cohomology, and cohomological dimension, of
monoids include [AR67, GP96, Nic69, Nic72, CS80, Nun95, GP98] and [Nov98]. The cohomological dimension of left regular bands was recently considered in [MSS15b] and [MSS15a] where
connections with the Leray number of simplicial complexes [KM05] and the homology of cell
complexes was obtained.
It is often easier to establish the topological finiteness properties Fn for a group than the
homological finiteness properties FPn , especially if there is a suitable geometry or topological
space available on which the group acts cocompactly. The desired homological finiteness properties can then be derived by the above-mentioned result for groups, that Fn (for n ≥ 2) is
equivalent to being finitely presented and of type FPn . In contrast, no corresponding theory of
Fn for monoids currently exists. Similarly, there is currently no analogue of geometric dimension of monoids in the literature. The study of homological finiteness properties of monoids
should greatly profit from the development of a corresponding theory of topological finiteness
properties of monoids. The central aim of the present article is to lay the foundations of such
a theory.
For such a theory to be useful in the study of homological finiteness properties of monoids
there are certain properties that any definition of left-Fn , and left geometric dimension, should
certainly satisfy. Specifically left-Fn should imply left-FPn , and the left geometric dimension
should provide an upper bound for the left cohomological dimension. The fundamental question
that needs to be addressed when developing this theory is to determine the correct analogue
of the K(G, 1)-complex in the theory for monoids? There is a natural notion of classifying
space |BM | of a monoid M . This is obtained by viewing M as a one-point category, letting
BM denote the nerve of this category, and setting |BM | as the geometric realisation of the
nerve; see Section 5 for full details of this construction. For a group G this space |BG| is a
K(G, 1)-complex, it is the canonical complex for G mentioned earlier in the introduction. Since
K(G, 1)s are unique up to homotopy equivalence the finiteness conditions Fn and cohomological
dimension can all be defined in terms of existence of CW complexes homotopy equivalent to

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ROBERT D. GRAY AND BENJAMIN STEINBERG

|BG| satisfying the appropriate finiteness property. Indeed in group theory it is a common
approach in the study of these topological finiteness properties to begin with the space |BG|
and then seek transformations on the space which preserve the homotopy equivalence class, but
make the space smaller. This is, for example, the basic idea behind the theory of collapsing
schemes (which will be discussed in more detail in Section 8). It could be regarded as natural
therefore to try define and study topological finiteness properties of a monoid M in terms of
the space |BM |. We note that there is an extensive literature on the study of classifying spaces
|BM | of monoids and related topics; see for instance [Fie84,Seg78,Nun95,McC69,McD79,MS76,
LN01, Pup58, Pup59, KT76, Hur89].
It turns out, however, that using |BM | to define topological finiteness properties of monoids
is not the right approach in the sense that it would lead to a definition of Fn for monoids which
does not imply left- or right-FPn , and there are similar issues for the corresponding definition
of geometric dimension. This essentially comes does to the fact that the space |BM | does not
contain enough information about the monoid M to recover the corresponding homological
finiteness properties.
In more detail, by applying results of MacDuff [McD79] it is possible to show that there are
examples of monoids which are not of type left-FP1 even though |BM | is contractible. For
example, if M is an infinite left zero semigroup (a semigroup with multiplication xy = x for
all elements x and y) with an identity adjoined then by [McD79, Lemma 5] the space |BM |
is contractible while it is straightforward to show that M does not even satisfy the property
left-FP1 (this also follows from Theorem 6.13 below). This shows that one should not define
property Fn for monoids using the space |BM |. Similar comments apply to attempts to define
geometric dimension–if one tries to define geometric dimension using |BM | then if M is any
monoid with a left zero element but no right zero element, the left cd(M ) would not equal zero
(by Proposition 6.28) while the geometric dimension would be zero.
This issue in fact arose in work of Brown [Bro92] when he introduced the theory of collapsing
schemes. In that paper Brown shows that if a monoid M admits a presentation by a finite
complete rewriting system, then |BM | has the homotopy type of a CW complex with only
finitely many cells in each dimension. When M is a group this automatically implies that the
group is of type FP∞ . Brown goes on to comment
“We would like, more generally, to construct a ‘small’ resolution of this type
for any monoid M with a good set of normal forms, not just for groups. I do
not know any way to formally deduce such a resolution from the existence of the
homotopy equivalence for |BM | above”.
As the comments above show, just knowing about the homotopy equivalence class of |BM |
will never suffice in order to deduce that the monoid is left- (or right-) FP∞ . It turns out
that the correct framework for studying topological finiteness properties of monoids is to pass
−−→
to the universal bundle |EM | over |BM |, which has a concrete description as the geometric
realisation of the right Cayley graph category of the monoid (this will be explained in detail
−−→
in Section 5). The space |EM | is contractable and the monoid has a natural action by left
multiplication on this space. This action is free and sends n-cells to n-cells. It turns out that
this space is the correct canonical model of what we shall call a left equivariant classifying space
for the monoid. In fact we are able to work in the more general context of projective M -sets,
and shall define a left equivariant classifying space as a contractable projective M -CW complex
(see Section 2 for the definition of projective M -CW complex). The corresponding finiteness
conditions left-Fn and left geometric dimension are then defined in the obvious natural way
in terms of the existence of a left equivariant classifying space satisfying appropriate finiteness
properties. Consequently, in order to define and study the topological finiteness properties of

TOPOLOGICAL FINITENESS PROPERTIES

5

monoids we are interested in, it will first be necessary for us to develop some of the foundations
of M -equivariant homotopy theory. Equivariant homotopy theory, and cohomology theory, for
groups is an important and well-established area; see for example [May96] for an introduction.
In this way, we are interested in studying homological finiteness properties of monoids by
investigating their actions on CW complexes.
The paper is structured as follows. The notions of free and projective M -CW complexes
are introduced in Section 2. For projective M -CW complexes we then go on to prove an
M -equivariant version of HELP (homotopy extension and lifting property), from which we
deduce the M -equivariant Whitehead theorems. We also apply HELP to prove that every
M -equivariant continuous mapping of projective M -CW complexes is M -homotopy equivalent
to a cellular one (the cellular approximation theorem). In Section 3 some useful base change
theorems are established. Section 4 is concerned with monoids acting on simplicial sets. In
particular we show how (rigid) M -CW complexes arise from (rigid) M -simplicial sets by taking
the geometric realisation. In Section 5 we recall the definition of the nerve of a category. By
considering three natural categories associated with a monoid M , namely the one-point category, the right Cayley graph category, and the two-sided Cayley graph category, we associate
−−→
←−→
three CW complexes with the monoid, denoted |BM |, |EM | and |EM |. We go through these
−−→
←−→
constructions in detail in that section. In particular the spaces |EM | and |EM | are the canonical models of M -equivariant (respectively two-sided M -equivariant) classifying spaces of the
monoid. In Section 6 we will define left-equivariant, and dually right-equivariant, classifying
spaces for a monoid M . We prove that such spaces always exist, and that they are unique
up to M -homotopy equivalence. Using the notion of left-equivariant classifying space we define property left-Fn for monoids. We prove several fundamental results about this property,
including results showing its relationship with property left-FPn , and its connection with the
properties of being finitely generated and finitely presented. We prove some results about the
closure properties of property left-Fn , in particular results relating the property holding in M
to it holding in certain maximal subgroups of M . We also introduce the notion of left geometric
dimension of a monoid in this section. We show that it is possible to find an M -finite equivariant classifying space for M which is projective when no M -finite free equivariant classifying
space exists, justifying our choice to work with projective M -CW complexes. The geometric dimension is proved to provide an upper bound for the cohomological dimension, and we
characterize monoids with left geometric dimension equal to zero. In Section 7 we introduce
the bilateral notion of a classifying space in order to introduce the stronger property of bi-Fn .
We prove results for bi-Fn analogous to those previously established for left- and right-Fn . In
particular we show that bi-Fn implies bi-FPn which is of interest from the point of view of
Hochschild cohomology. We also define the geometric dimension as the minimum dimension of
a bi-equivariant classifying space and show how it relates to the Hochschild dimension. In Sections 8 and 9 we develop the theory of M -equivariant collapsing schemes [Bro92]. Equivalently,
this may be viewed as the development of M -equivariant discrete Morse theory in the sense
of Forman [For02]. We show in Section 10 that this theory can be applied to monoids which
admit a, so-called, guarded collapsing scheme. Then, in Section 11, we identify some classes of
monoids which admit guarded collapsing schemes, and in particular recover a topological proof
of the Anick’s theorem, and more generally Kobayashi’s theorem, that monoids defined by finite
complete rewriting systems are of type bi-FP∞ (by proving that they are all of type bi-F∞ ).
Brown proved Anick’s theorem by developing a theory of collapsing schemes (or discrete Morse
theory) for chain complexes, which has been rediscovered by other authors [Koz05] later on.
Our approach obviates the need to develop a chain complex analogue in this setting.

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ROBERT D. GRAY AND BENJAMIN STEINBERG

Applications of the topological approach set out in this paper will be given in a future
article by the current authors [GS]. Among other things in that paper we shall show how
our topological approach can be used to prove results about the closure properties of left-Fn
and bi-Fn for (i) amalgamated free products of monoids (simplifying and vastly improving on
some results of Cremanns and Otto [CO98]) (ii) HNN-like extensions in the sense of Otto and
Pride [PO04] (in particular establishing results which generalise [PO04, Theorem 1] to higher
dimensions), and (iii) HNN extensions of the sort considered by Howie [How63]. For example,
we prove that if A, B are monoids containing a common submonoid C such that A and B are
free right C-set, then if A and B are of type left-Fn and C is of type left-Fn−1 , then A ∗C B is of
type left-Fn . An analogous result is proved for the homological finiteness property FPn , under
the weaker hypothesis that ZA and ZB are flat as ZC-modules. Monoid amalgamated products
are much more complicated than group ones. For instance, the amalgamated free product of
finite monoids can have an undecidable word problem, and the factors do not have to embed
or intersect in the base monoid.
Additionally, we shall give a topological proof that a free inverse monoid on one or more
generators is neither of type left-FP2 nor right-FP2 generalising a classical result of Schein
[Sch75] that such monoids are not finitely presented.
Finally, in [GS] we shall apply our methods to prove results about the homological finiteness
properties of special monoids, that is, monoids defined by finite presentations of the form
hA | w1 = 1, . . . , wk = 1i. We prove that if M is a special monoid with group of units G then
if G is of type FPn with 1 ≤ n ≤ ∞, then M is of type left- and right-FPn ; and moreover that
cd G ≤ cd M ≤ max{2, cd G}. As a corollary we obtain that all special one-relation monoids are
of type left- and right-FP∞ , answering a special case of a question of Kobayashi [Kob00], and
we recover Kobayashi’s result that if the relator is not a proper power then the cohomological
dimension is at most 2. Specifically we show that if M is a special one-relation monoid then
M is of type left- and right-FP∞ , and if M = hA | w = 1i with w not a proper power, then
cd M ≤ 2; otherwise cd M = ∞.
2. Projective M -CW complexes and M -homotopy theory
2.1. CW complexes and topological finiteness properties of groups. For background
on CW complexes, homotopy theory, and algebraic topology for group theory, we refer the
reader to [Geo08] and [May99]. Throughout B n will denote the closed unit ball in Rn , S n−1
the (n − 1)-sphere which is the boundary ∂B n of the n-ball, and I = [0, 1] the unit interval. We
use en to denote an open n-cell, homeomorphic to the open n ball B̊ n = B n − ∂B n , ∂e denotes
the boundary of e and ē = cl(e) the closure of e, respectively. We identify I r = I r × 0 ⊂ I r+1 .
A CW complex is a space X which is a union of subspaces Xn such that, inductively, X0 is
a discrete set of points, and Xn is obtained from Xn−1 by attaching balls B n along attaching
maps j : S n−1 → Xn−1 . The resulting maps B n → Xn are called the characteristic maps. So
Xn is the quotient space obtained from Xn−1 ∪ (Jn × Bn ) by identifying (j, x) with j(x) for
x ∈ S n , where Jn is the discrete set of such attaching maps. Thus Xn is obtained as a pushout
of spaces:
Jn × S n−1

Xn−1

Jn × B n

Xn .

The topology of X should be that of the inductive limit X = lim Xn
−→

TOPOLOGICAL FINITENESS PROPERTIES

7

S
A CW complex X is then equal as a set to the disjoint union of (open) cells X = α eα
where the eα are the images of B̊ n under the characteristic maps. Indeed, an alternative way of
defining CW complex, which shall be useful for us to use later on, is as follows. A CW complex
is a Hausdorff space K along with a family {eα } of open cells of various dimensions such that,
letting
[
K j = {eα : dim eα ≤ j},

the following conditions are satisfied
[
eα and eα ∩ eβ = ∅ for α 6= β.
(CW1) K =
α

(CW2) For each cell eα there is a map ϕα : B n → K (called the characteristic map) where B n
is a topological ball of dimension n = dim eα , such that
(a) ϕα |B̊ n is a homeomorphism onto eα ;
(b) ϕα (∂B n ) ⊂ K n−1 .
(CW3) Each eα0 is contained in a union of finitely many eα .
(CW4) A set A ⊂ K is closed in K if and only if A ∩ eα is closed in eα for all eα .
Note that each characteristic map ϕ : B n → K gives rise to a characteristic map ϕ′ : I n → K
be setting ϕ′ = ϕh for some homeomorphism h : I n → B n . So we can restrict our attention to
characteristic maps with domain I n when convenient. If ϕ : B n → K is a characteristic map
map for e. A subcomplex is a subset L ⊂ K with a
for a cell e then ϕ|∂B n is called an attaching
[
subfamily {eβ } of cells such that L =
eβ and every eβ is contained in L. If L is a subcomplex
of K we write L < K and call (K, L) a CW pair. If e is a cell of K which does not lie in (and
hence does not meet) L we write e ∈ K − L. An isomorphism between CW complexes is a
homeomorphism that maps cells to cells.
Let M be a monoid. We shall define notions of free and projective M -CW complexes and
then use these to study topological finiteness properties of M . The notion of a free M -CW
complex is a special case of a free C-CW complex for a category C considered by Davis and
Lück in [DL98] and so the cellular approximation theorem, HELP Theorem and Whitehead
Theorem in this case can be deduced from their results. The HELP Theorem and Whitehead
Theorem for for projective M -CW complexes can be extracted with some work from the more
general results of Farjoun [DFZ86] on diagrams of spaces but to keep things elementary and
self-contained we present them here .
2.2. The category of M -sets. A left M -set consists of a set X and a mapping M × X → X
written (m, x) 7→ mx called a left action, such that 1x = x and m(nx) = (mn)x for all m, n ∈ M
and x ∈ X. Right M -sets are defined dually, they are the same thing as left M op -sets. A biM -set is an M × M op -set. There is a category of M -sets and M -equivariant mappings, where
f : X → Y is M -equivariant if f (mx) = mf (x) for all x ∈ X, m ∈ M .
A (left) M -set X is said to be free on a set A if there is a mapping ι : A → X such that for
any mapping f : A → Y with Y an M -set, there is a unique M -equivariant map F : X → Y
such that
A ι X
F

f

Y

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ROBERT D. GRAY AND BENJAMIN STEINBERG

commutes. The mapping ι is necessarily injective. If X is an M -set and A ⊆ X, then A is a
free basis for X if and only if each element of X can be uniquely expressed as ma with m ∈ M
and a ∈ A.
The free left M -set on A exists and can be realised as the set M × A with action m(m′ , a) =
(mm′ , a) and ι is the map a 7→ (1, a). Note that if G is a group, then a left G-set X is free if
and only if G acts freely on X, that is, each element of X has trivial stabilizer. In this case,
any set of orbit representatives is a basis.
An M -set P is projective if any M -equivariant surjective mapping f : X → P has an M equivariant section s : P → X with f ◦ s = 1P . Free M -sets are projective and an M -set is
projective if and only if it is a retract of a free one.
`
Each projective M -set P is isomorphic to an M -set of the form a∈A M ea (disjoint union,
which is the coproduct in the category of M -sets) with ea ∈ E(M ). Here E(M ) denotes the
set of idempotents of the monoid M . In particular, projective G-sets are the same thing as free
G-sets for a group G. (See [Kna72] for more details.)
2.3. Equivariant CW complexes. A left M -space is a topological space X with a continuous
left action M × X → X where M has the discrete topology. A right M -space is the same thing
as an M op -space and a bi-M -space is an M × M op -space. Each M -set can be viewed as a
discrete M -space. Note that colimits in the category of M -spaces are formed by taking colimits
in the category of spaces and observing that the result has a natural M -action.
Let us define a (projective) M -cell of dimension n to be an M -space of the form M e × B n
where e ∈ E(M ) and B n has the trivial action; if e = 1, we call it a free M -cell. We will define
a projective M -CW complex in an inductive fashion by imitating the usual definition of a CW
complex but by attaching M -cells M e × B n via M -equivariant maps from M e × S n−1 to the
(n − 1)-skeleton.
Formally, a projective (left) relative M -CW complex is a pair (X, A) of M -spaces such that
X = lim Xn with in : Xn → Xn+1 inclusions, X−1 = A, X0 = P0 ∪ A with P0 a projective M -set
−→
and where Xn is obtained as a pushout of M -spaces
Pn × S n−1

Xn−1
(2.1)

Pn × B n

Xn

with Pn a projective M -set and B n having a trivial M -action for n ≥ 1. As usual, Xn is called
the n-skeleton of X and if Xn = X and Pn 6= ∅, then X is said to have dimension n. Notice
that since Pn is isomorphic to a coproduct of M -sets of the form M e with e ∈ E(M ), we are
indeed attaching M -cells at each step. If A = ∅, we call X a projective M -CW complex. Note
that a projective M -CW complex is a CW complex and the M -action is cellular (in fact, takes
n-cells to n-cells). We can define projective right M -CW complexes and projective bi-M -CW
complexes by replacing M with M op and M × M op , respectively. We say that X is a free
M -CW complex if each Pn is a free M -set. If G is a group, a CW complex with a G-action is a
free G-CW complex if and only if G acts freely, cellularly, taking cells to cells, and the setwise
stabilizer of each cell is trivial [Geo08, Appendix of Section 4.1].
More generally we define an M -CW complex in the same way as above except that the
Pi are allowed to be arbitrary M -sets. Most of the theory developed below is only valid in
the projective setting, but there will be a few occasions (e.g. when we discuss M -simplicial
sets) where it will be useful for us to be able to refer to M -CW complexes in general. For
future reference we should note here that, just as for the theory of G-CW complexes, there
is alternative way of defining M -CW complex in terms of monoids acting on CW complexes.

TOPOLOGICAL FINITENESS PROPERTIES

9

This follows the same lines as that of groups, see for example [Geo08, Section
3.2 and page 110]
S
or [May96]. Let Y be a left M -space where M is a monoid and Y = α eα is a CW complex
with characteristic maps ϕα : B n → Y . We say that Y is a rigid left M -CW complex if it is:
• Cellular and dimension preserving: For every eα and m ∈ M there exists an eβ such that
meα = eβ and dim(eβ ) = dim(eα ); and
• Rigid on cells: If meα = eβ then mϕα (k′ ) = ϕβ (k′ ) for all k′ ∈ B n − ∂B n .
If the action of M on the set of n-cells is free (respectively projective) then we call Y a free
(respectively projective) rigid left M -CW complex. The inductive process described above for
building (projective, free) left M -CW complexes is easily seen to give rise to rigid (projective,
free) left M -CW complexes, in the above sense. Conversely every rigid (projective, free) left
M -CW complex arises in this way. In other words, the two definitions are equivalent. For an
explanation of this in the case of G-CW complexes see, for example, [Geo08, page 110]. The
proof for monoids is analogous and is omitted. Similar comments apply for rigid right M -CW
complexes and rigid bi-M -CW complexes.
We say that a projective M -CW complex X is of M -finite type if Pn is a finitely generated
projective M -set for each n and we say that X is M -finite if it is finite dimensional and of
M -finite type (i.e., X is constructed from finitely many M -cells).
Notice that if m ∈ M , then mX is a subcomplex of X for all m ∈ M with n-skeleton mXn .
Indeed, mX0 = mP0 is a discrete set of points and mXn is obtained from mXn−1 via the
pushout diagam
mPn × S n−1

mXn−1

mPn × B n

mXn .

A projective M -CW subcomplex of X is an M -invariant subcomplex A ⊆ X which
` ′′is a union
′
of M -cells of X. In other words, each Pn (as above) can be written Pn = Pn Pn with the
images of the Pn′ × B n giving the cells of A. Notice that if A is a projective M -CW subcomplex
of X, then (X, A) can be viewed as a projective relative M -CW complex in a natural way. Also
note that a cell of X belongs to A if and only if each of its translates do.
A projective {1}-CW complex is the same thing as a CW complex and {1}-finite type ({1}finite) is the same thing as finite type (finite).
If e ∈ E(M ) is an idempotent and m ∈ M e, then left multiplication by m induces an
isomorphism Hn ({e} × B n , {e} × S n−1 ) → Hn ({m} × B n , {m} × S n−1 ) (since it induces a
homeomorphism {e} × B n /{e} × S n−1 → {m} × B n /{m} × S n−1 ) and so if we choose an
orientation for the n-cell {e} × B n , then we can give {m} × B n the orientation induced by this
isomorphism. If m ∈ M and m′ ∈ M e, then the isomorphism
Hn ({e} × B n , {e} × S n−1 ) → Hn ({mm′ } × B n , {mm′ } × S n−1 )
induced by mm′ is the composition of the isomorphism
Hn ({e} × B n , {e} × S n−1 ) → Hn ({m′ } × B n , {m′ } × S n−1 )
induced by m′ and the isomorphism
Hn ({m′ } × B n , {m′ } × S n−1 ) → Hn ({mm′ } × B n , {mm′ } × S n−1 )
induced by m and so the action of m preserves orientation. We conclude that the degree n
component of the cellular chain complex for X is isomorphic to ZPn as a ZM -module and

10

ROBERT D. GRAY AND BENJAMIN STEINBERG

 L
`
∼
∼
hence is projective (since Z
a∈A ZM ea and ZM = ZM e ⊕ ZM (1 − e) for any
a∈A M ea =
idempotent e ∈ E(M )).
If X is a projective M -CW complex then so is Y = M × I `
where I is given the trivial action.
∼
If we retain the above notation, then Y0 = X`
×
∂I
X
X0 . The n-cells for n ≥ 1 are
=
0
0
n and P
n−1 × I. Notice that
obtained from attaching Pn × B n × ∂I ∼
(P
P
)
×
B
= n
n
n−1 × B
X × ∂I is a projective M -CW subcomplex of X × I.
If X, Y are M -spaces, then an M -homotopy between M -equivariant continuous maps f, g : X →
Y is an M -equivariant mapping H : X × I → Y with H(x, 0) = f (x) and H(x, 1) = g(x) for
x ∈ X where I is viewed as having the trivial M -action. We write f ≃M g in this case. We say
that X, Y are M -homotopy equivalent, written X ≃M Y , if there are M -equivariant continuous
mappings (called M -homotopy equivalences) f : X → Y and g : Y → X with gf ≃M 1X and
f g ≃M 1Y . We write [X, Y ]M for the set of M -homotopy classes of M -equivariant continuous
mappings X → Y .
Lemma 2.1. Let X, Y be projective M -CW complexes and A a projective M -CW subcomplex
`
of X. Let f : A → Y be a continuous M -equivariant cellular map. Then the pushout X A Y
is a projective M -CW complex.
`
Proof. It is a standard result that X A Y is a CW complex whose n-cells are the`n-cells of Y
together with the n-cells of X not belonging to A. In more detail, let q : X → X A Y be the
canonical mapping and view Y as a subspace of the adjunction space. Then the attaching map
of a cell coming from Y is the original attaching map, whereas the attaching map of a cell of
X not belonging to A is the composition of q with its original attaching mapping. `
It follows
from the definition of a projective M -CW subcomplex and the construction that X A Y is a
projective M -CW complex. Here it is important that a translate by M of a cell from X \ A is
a cell of X \ A.

A free M -CW subcomplex of a free M -CW complex X is an M -invariant subcomplex A ⊆ X
which is a union of M -cells of X.
The proof of Lemma 2.1 yields the following.
Lemma 2.2. Let X, Y be free M -CW complexes and A a free M -CW subcomplex
` of X. Let
f : A → Y be a continuous M -equivariant cellular map. Then the pushout X A Y is a free
M -CW complex.
A continuous mapping f : X → Y of spaces is an n-equivalence if
f∗ : πq (X, x) → πq (Y, f (x))
is a bijection for 0 ≤ q < n and a surjection for q = n where π0 (Z, z) = π0 (Z) (viewed as a
pointed set with base point the component of z). It is a weak equivalence if it is an n-equivalence
for all n, i.e., f∗ is a bijection for all q ≥ 0. We will consider a weak equivalence as an ∞equivalence. We shall see later that an M -equivariant weak equivalence of projective M -CW
complexes is an M -homotopy equivalence.
Let Top(X, Y ) denote the set of continuous maps X → Y for spaces X, Y and TopM (X, Y )
denote the set of continuous M -equivariant maps X → Y between M -spaces X, Y .
Proposition 2.3. Let X be a space with a trivial M -action, e ∈ E(M ) and Y an M -space.
Then there is a bijection between TopM (M e × X, Y ) and Top(X, eY ). The bijection sends
g: Me × X → Y
f : M e × X → Y to f : X → eY given by f (x) = f (e, x) and g : X → eY to b
given by gb(m, x) = mg(x).

TOPOLOGICAL FINITENESS PROPERTIES

11

Proof. If x ∈ X, then f (x) = f (e, x) = f (e(e, x)) = ef (e, x) ∈ eY . Clearly, f is continuous. As
gb is the composition of 1M e × g with the action map, it follows that gb is continuous. We show
that the two constructions are mutually inverse. First we check that
fb(m, x) = mf (x) = mf (e, x) = f (m(e, x)) = f (me, x) = f (m, x)

for m ∈ M e and x ∈ X. Next we compute that

gb(x) = gb(e, x) = eg(x) = g(x)

since g(x) ∈ eY . This completes the proof.



Proposition 2.3 is the key tool to transform statements about projective M -CW complexes
into statement about CW complexes. We shall also need the following lemma relating equivariant n-equivalences and n-equivalences.
Lemma 2.4. Let Y, Z be M -spaces and let k : Y → Z be an M -equivariant n-equivalence with
0 ≤ n ≤ ∞. Let e ∈ E(M ) and k′ = k|eY : eY → eZ. Then k′ is an n-equivalence.
Proof. First note that k(ey) = ek(y) and so k|eY does indeed have image contained in eZ. Let
y ∈ eY and q ≥ 0. Let α : eY → Y and β : eZ → Z be the inclusions. Then note that the
action of e gives retractions Y → eY and Z → eZ. Hence we have a commutative diagram
k∗

πq (Y, y)
α∗

πq (Z, k(y))

e∗

β∗

πq (eY, y)

k∗′

e∗

πq (eZ, k(y))

with e∗ α∗ and e∗ β∗ identities. Therefore, if k∗ is surjective, then k∗′ is surjective and if k∗ is

injective, then k∗′ is injective. The lemma follows.
2.4. Whitehead’s theorem. With Lemma 2.4 in hand, we can prove an M -equivariant version
of HELP (homotopy extension and lifting property) [May99, Page 75], which underlies most of
the usual homotopy theoretic results about CW complexes. If X is a space, then ij : X → X ×I,
for j = 0, 1, is defined by ij (x) = (x, j).
Theorem 2.5 (HELP). Let (X, A) be a projective relative M -CW complex of dimension at
most n ∈ N ∪ {∞} and k : Y → Z an M -equivariant n-equivalence of M -spaces. Then given M equivariant continuous mappings f : X → Z, g : A → Y and h : A × I → Z such that kg = hi1
and f i = hi0 (where i : A → X is the inclusion), there exist M -equivariant continuous mappings
ge and e
h making the diagram
i0

A

A×I

i1

A

h

g

Y

e
h

f

X

k

Z

i

i0

i
g
e

X ×I

i1

X

commute.
Proof. Proceeding by induction on the skeleta and adjoining an M -cell at a time, it suffices to
handle the case that
(X, A) = (M e × B q , M e × S q−1 )

12

ROBERT D. GRAY AND BENJAMIN STEINBERG

with 0 ≤ q ≤ n. By Proposition 2.3 it suffices to find continuous mappings ge and e
h making the
diagram
i0

S q−1

S q−1 × I

i1

S q−1

h
g

eY

e
h

f

Bq

k

eZ

i

i0

i
g
e

Bq × I

i1

Bq

commute where we have retained the notation of Proposition 2.3. The mapping k : eY → eZ is
an n-equivalence by Lemma 2.4 and so we can apply the usual HELP theorem [May99, Page 75]
for CW complexes to deduce the existence of ge and e
h. This completes the proof.

As a consequence we may deduce the M -equivariant Whitehead theorems.

Theorem 2.6 (Whitehead). If X is a projective M -CW complex and k : Y → Z is an M equivariant n-equivalence of M -spaces, then the induced mapping k∗ : [X, Y ]M → [X, Z]M is a
bijection if dim X < n or n = ∞ and a surjection if dim X = n < ∞.
Proof. For surjectivity we apply Theorem 2.5 to the pair (X, ∅). If f : X → Z, then e
g: X → Y
satisfies kg ≃M f . For injectivity, we apply Theorem 2.5 to the pair (X × I, X × ∂I) and note
that X ×I has dimension one larger than X.
` Suppose that p, q : X → Y are such that kp ≃M kq
via a homotopy f : X × I → Z. Put g = p q : X × ∂I → Y and define h : X × ∂I × I → Z by
h(x, s, t) = f (x, s). Then e
g : X × I → Y is a homotopy between p and q.


Corollary 2.7 (Whitehead). If k : Y → Z an M -equivariant weak equivalence (n-equivalence)
between projective M -CW complexes (of dimension less than n), then k is an M -homotopy
equivalence.
Proof. Under either hypothesis, k∗ : [Z, Y ]M → [Z, Z]M is a bijection by Theorem 2.6 and so
kg ≃M 1Z for some M -equivariant g : Z → Y . Then kgk ≃M k and hence, since k∗ : [Y, Y ] →
[Y, Z] is a bijection by Theorem 2.6, we have that gk ≃M 1Y . This completes the proof.

2.5. Cellular approximation. Our next goal is to show that every M -equivariant continuous
mapping of projective M -CW complexes is M -homotopy equivalent to a cellular one. We
shall need the well-known fact that if Y is a CW complex, then the inclusion Yn ֒→ Y is an
n-equivalence for all n ≥ 0 [May99, Page 76].
Theorem 2.8 (Cellular approximation). Let f : X → Y be a continuous M -equivariant mapping with X a projective M -CW complex and Y a CW complex with a continuous action of M
by cellular mappings. Then f is M -homotopic to a continuous M -equivariant cellular mapping.
Any two cellular approximations are homotopy equivalent via a cellular M -homotopy.
Proof. We prove only the first statement. The second is proved using a relative version of the
first whose statement and proof we omit. Note that Yn is M -equivariant for all n ≥ 0 because
M acts by cellular mappings. We construct by induction M -equivariant continuous mappings
fn : Xn → Yn such that f |Xn ≃M fn |Xn via an M -homotopy hn and fn , hn extend f`n−1 , hn−1 ,
respectively (where we take X−1 = ∅). We have, without loss of generality, X0 = a∈A M ea .
Since ea Y is a subcomplex of Y with 0-skeleton ea Y0 and f (ea ) ∈ ea Y , we can find a path pa
in ea Y from f (ea ) to an element ya ∈ ea Y0 . Define f0 (mea ) = mya and h0 (mea , t) = mpa (t),
cf. Proposition 2.3.

TOPOLOGICAL FINITENESS PROPERTIES

13

Assume now that fn , hn have been defined. Since the inclusion Yn+1 → Y is an M -equivariant
(n + 1)-equivalence, Theorem 2.5 gives a commutative diagram
i0

Xn

Xn × I

i1

Xn

hn
fn

Y

i

Yn+1
hn+1

f

Xn+1

i0

i
fn+1

Xn+1 × I

i1

Xn+1

thereby establishing the inductive step. We obtain our desired cellular mapping and M homotopy by taking the colimit of the fn and hn .

3. Base change
If A is a right M -set and B is a left M -set, then A ⊗M B is the quotient of A × B by the least
equivalence relation ∼ such that (am, b) ∼ (a, mb) for all a ∈ A, b ∈ B and m ∈ M . We write
a ⊗ b for the class of (a, b) and note that the mapping (a, b) 7→ a ⊗ b is universal for mappings
f : A × B → X with X a set and f (am, b) = f (a, mb). If M happens to be a group, then M
acts on A × B via m(a, b) = (am−1 , mb) and A ⊗M B is just the set of orbits of this action.
The tensor product A ⊗M () preserves all colimits because it is a left adjoint to the functor
X 7→ X A .
If B is a left M -set there is a natural preorder relation ≤ on B where x ≤ y if and only if
M x ⊆ M y. Let ≈ denote the symmetric-transitive closure of ≤. That is, x ≈ y if there is a
sequence z1 , z2 , . . . , zn of elements of B such that for each 0 ≤ i ≤ n − 1 either zi ≤ zi+1 or
zi ≥ zi+1 . This is clearly an equivalence relation and we call the ≈-classes of B the weak orbits
of the M -set. This corresponds to the notion of the weakly connected components of a directed
graph. If B is a right M -set then we use B/M to denote the set of weak orbits of the M -set.
Dually, if B is a left M -set we use M \B to denote the set of weak orbits. Note that if 1 denotes
the trivial right M -set and B is a left M -set, then we have M \B = 1 ⊗M B.
Let M, N be monoids. An M -N -biset is an M × N op -set. If A is an M -N -biset and B is a
left N -set, then the equivalence relation defining A ⊗N B is left M -invariant and so A ⊗N B is
a left M -set with action m(a ⊗ b) = ma ⊗ b.
Proposition 3.1. Let A be an M -N -biset that is (finitely generated) projective as an M -set
and let B be a (finitely generated) projective N -set. Then A ⊗N B is a (finitely generated)
projective M -set.
Proof. As B is a (finite) coproduct of N -sets N e with e ∈ E(N ), it suffices to handle the case
B = N e. Then A ⊗N N e ∼
= Ae via a ⊗ n 7→ an with inverse a 7→ a ⊗ e for a ∈ Ae. Now
define r : A → Ae by r(a) = ae. Then r is an M -equivariant retraction. So Ae is a retract of a
(finitely generated) projective and hence is a (finitely generated) projective.

If X is a left M -space and A is a right M -set, then A ⊗M X is a topological space with the
quotient topology. Again the functor A ⊗M () preserves all colimits. In fact, A ⊗M X is the
coequalizer in the diagram
a
a
X⇒
X → A ⊗M X
A×M

A

where the top map sends x in the (a, m)-component to mx in the a-component and the bottom
map sends x in the (a, m)-component to x in the am-component.

14

ROBERT D. GRAY AND BENJAMIN STEINBERG

Corollary 3.2. If A is an M -N -biset that is projective as an M -set and X is a projective
N -CW complex, then A ⊗N X is a projective M -CW complex. If A is in addition finitely
generated as an M -set and X is of N -finite type, then A ⊗N X is of M -finite type. Moreover,
dim A ⊗N X = dim X.
Proof. Since A ⊗N () preserves colimits, A ⊗N X = lim A ⊗N Xn . Moreover, putting X−1 = ∅,
−→
we have that if Xn is obtained as per the pushout square (2.1), then A ⊗N Xn is obtained from
the pushout square
(A ⊗N Pn ) × S n−1

A ⊗N Xn−1

(A ⊗N Pn ) × B n

A ⊗N Xn

by preservation of colimits and the observation that if C is a trivial left N -set and B is a left
N -set, then A ⊗N (B × C) ∼
= (A ⊗N B) × C via a ⊗ (b, c) 7→ (a ⊗ b, c). The result now follows
from Proposition 3.1

By considering the special case where M is trivial and A is a singleton, and observing that
a projective M -set P is finitely generated if and only if M \P is finite, we obtain the following
corollary.
Corollary 3.3. Let X be a projective M -CW complex. Then M \X is a CW complex. Moreover,
X is M -finite (of M -finite type) if and only if M \X is finite (of finite type).
The following observation will be used many times.
Proposition 3.4. Let X be a locally path connected N -space and A an M -N -biset. Then π0 (X)
is an N -set and π0 (A ⊗N X) ∼
= A ⊗N π0 (X).
Proof. Note that the functor X 7→ π0 (X) is left adjoint to the inclusion of the category of N -sets
into the category of locally path connected M -spaces and hence it preserves all colimits. The
result now follows from the description of tensor products as coequalizers of coproducts.

The advantage of working with M -homotopies is that they behave well under base change.
Proposition 3.5. Let A be an M -N -biset and let X, X ′ be N -homotopy equivalent N -spaces.
Then A ⊗N X ′ is M -homotopy equivalent to A ⊗N X ′ . In particular, if Y, Z are M -spaces and
Y ≃M Z, then M \Y ≃ M \Z.
Proof. It suffices to prove that if Y, Z are N -spaces and f, g : Y → Z are N -homotopic N equivariant maps, then
A ⊗N f, A ⊗N g : A ⊗N Y → A ⊗N Z
are M -homotopic. This follows immediately from the identification of A ⊗N (Y × I) with
(A ⊗N Y ) × I. For if H : Y × I → Z is an N -homotopy between f and g, then A ⊗N H provides
the M -homotopy between A ⊗N f and A ⊗N g.

The following base change lemma, and its dual, is convenient for dealing with bisets.
Lemma 3.6. Let A be an M × M op -set and consider the right M × M op -set M with the
right action m(mL , mR ) = mmL . Then A/M is a left M -set and there is an M -equivariant
isomorphism A/M → M ⊗M ×M op A.

TOPOLOGICAL FINITENESS PROPERTIES

15

Proof. Clearly, A/M = A ⊗M 1 is a left M -set. Write [a] for the class of a in A/M . Define
f : A/M → M ⊗M ×M op A by f ([a]) = 1 ⊗ a. This is well defined and M -equivariant because if
a ∈ A and m ∈ M , then 1 ⊗ am = 1 ⊗ (1, m)a = 1(1, m) ⊗ a = 1 ⊗ a and 1 ⊗ ma = 1 ⊗ (m, 1)a =
1(m, 1) ⊗ a = m ⊗ a. Define G : M × A → A/M by G(m, a) = [ma]. If m, mL , mR ∈ M , then
G(m(mL , mR ), a) = G(mmL , a) = [mmL a] and G(m, (mL , mR )a) = [mmL amR ] = [mmL a].
Therefore, G induces a well defined mapping g : M ⊗M ×M op A → A/M . Then we check that
gf ([a]) = g(1⊗ a) = [a] and f g(m ⊗ a) = f ([ma]) = 1⊗ ma = 1⊗ (m, 1)a = 1(m, 1)⊗ a = m ⊗ a.
Thus f and g are inverse isomorphisms.

The following basic result will be used later.
Proposition 3.7. Let G be a group. Then G × G is a (G × Gop )-G-biset that is free as right
G-set on cardinality of G generators under the right action (g, g ′ )h = (gh, h−1 g ′ ).
Proof. It is easy to check that right action of G is indeed an action commuting with the left
action of G × Gop . Moreover, the right action of G is free and two elements (g1 , g2 ) and (g1′ , g2′ )

are in the same right G-orbit if and only if g1 g2 = g1′ g2′ . This completes the proof.
Corollary 3.8. Let M be a monoid and X a projective M × M op -CW complex.
(1) X/M is a projective M -CW complex and M \X is a projective M op -CW complex.
(2) If X is of M × M op -finite type, then X/M is of M -finite type and dually for M \X.
(3) dim X/M = dim X = dim M \X.
(4) If X, Y are M × M op -homotopic projective M × M op -CW complexes, then X/M and
Y /M (respectively, M \X and M \Y ) are M -homotopic projective M -CW complexes
(respectively, M op -homotopic projective M op -CW complexes).
Proof. The first three items follow from Corollary 3.2 and Lemma 3.6 (and their duals). The
final statement follows from Lemma 3.6 and Proposition 3.5.

We shall frequently use without comment that if A is an M -N -biset and B is an N -set, then
Z[A ⊗N B] ∼
= ZA ⊗ZN ZB as left ZM -modules. Indeed, there are natural isomorphisms of
abelian groups
HomZM (ZA ⊗ZN ZB, V ) ∼
= HomZN (ZB, HomZM (ZA, V ))
∼
= HomN (B, HomM (A, V ))
∼
= HomM (A ⊗N B, V )
∼
= HomZM (Z[A ⊗N B], V )
for a ZM -module V and so we can apply Yoneda’s Lemma.
4. Simplicial sets
An important source of examples of rigid M -CW complexes will come from simplicial sets
which admit suitable monoid actions. In this section we introduce the notion of a rigid M simplicial set, and we show how these give rise to rigid M -CW complexes via the geometric
realisation functor. For further background on simplicial sets we refer the reader to [Wei94,
Chapter 8] or [May67].
Let ∆ denote the simplicial category. It has objects all the finite linearly ordered sets [n] =
{0, 1, . . . , n − 1} (n ≥ 0) and morphisms given by (non-strictly) order-preserving maps. A
simplicial set X is then a functor X : ∆op → Set from ∆op to the category of sets. For each n,
the image of [n] under X is denoted Xn and is called the set of n-simplicies of the simplicial

16

ROBERT D. GRAY AND BENJAMIN STEINBERG

set. Any simplicial set X may be defined combinatorially as a collection of sets Xn (n ≥ 0) and
functions di : Xn → Xn−1 and si : Xn → Xn+1 (0 ≤ i ≤ n) satisfying
di dj = dj−1 di

(i < j)

si sj = sj+1 si (i ≤ j)


i = j, j + 1
1
di sj = sj−1 di i < j


sj di−1 i > j + 1.

Here the di are called the face maps and the si are called the degeneracy maps. We say that
an n-simplex x ∈ Xn is degenerate if it is the image of some degeneracy map.
A simplicial morphism f : X → Y between simplicial sets is a natural transformation between
the corresponding functors, i.e., a sequence of functions fn : Xn → Yn for each n ≥ 0 such that
fn−1 di = di fn and fn+1 sj = sj fn . There is a functor | · | : SSet → CG, called the geometric
realization functor, from the category SSet of simplicial
sets and the category CG of compactlyS
generated Hausdorff topological spaces. Let K = i≥0 Ki be a simplicial set with degeneracy
and face maps di , si . The geometric realisation |K| of K is the CW complex constructed from
K in the following way. Let
n
o
X
∆n = (t0 , . . . , tn ) : 0 ≤ ti ≤ 1,
ti = 1 ⊆ Rn+1
denote the standard topological n-simplex. Define
δi :

∆n−1
→
∆n
(t0 , . . . , tn−1 ) 7→ (t0 , . . . , ti−1 , 0, ti , . . . , tn−1 ),

σi :

∆n+1
→
∆n
(t0 , . . . , tn+1 ) 7→ (t0 , . . . , ti + ti+1 , . . . , tn−1 ).

and

Then



|K| = 

G

n≥0



Kn × ∆n  /∼

where ∼ is the equivalence relation generated by
(x, δi (u)) ∼ (di (x), u),
We give




G

0≤n≤q

(x, σi (v)) ∼ (si (x), v).


Kn × ∆n  /∼

the quotient topology for all q and take the inductive limit of the resulting topologies. The
geometric realisation |K| is a CW complex whose cells are in natural bijective correspondence
with the non-degenerate simplicies of K. To see this, write
G
K=
Kn × ∆n .
n≥0

Then a point (k, x) ∈ K is called non-degenerate if k is a non-degenerate simplex and x is an
interior point. The following is [Mil57, Lemma 3].
Lemma 4.1. Each point (k, x) ∈ K is ∼-equivalent to a unique non-degenerate point.

TOPOLOGICAL FINITENESS PROPERTIES

17

In each case, the point in question is determined by the maps δi , di , σi and si (see [Mil57] for
details). This lemma is the key to proving that that |K| is a CW complex: we take as n-cells
of |K| the images of the non-degenerate n-simplices of K, and the above lemma shows that the
interiors of these cells partition |K|. The remaining properties of a CW complex are then easily
verified. The following lemma shows that geometric realisation defines a functor from SSet to
CG.
The next result is [Mil57, Lemma 4].
S
S
Lemma 4.2. If K = Ki and L = Li are simplicial sets and f : K → L is a simplicial
morphism then f given by
f n : Kn × ∆n → Ln × ∆n ,

(x, u) 7→ (f (x), u)

is continuous, and induces a well-defined continuous map
|f | : |K| → |L|,

(x, u)/∼ 7→ (f (x), u)/∼

of the corresponding geometric realizations, which is cellular.
A left M -simplicial set is a simplicial set equipped with a left action of M by simplicial
morphisms. In order to construct rigid M -CW complexes we shall need the following special
kind of M -simplicial set.
S
Definition 4.3 (Rigid M -simplicial set). Let K = i≥0 Ki be a simplicial set with degeneracy
and face maps di , si , and let M be a monoid. We call K a rigid left M -simplicial set if K
comes equipped with an action of M × K → K such that
• M is acting by simplicial morphisms, i.e., M maps n-simplicies to n-simplicies, and
commutes with di and si ;
• M preserves non-degeneracy, i.e., for every non-degenerate n-simplex x and every m ∈
M the n-simplex mx is also non-degenerate.
A rigid right M -simplicial set is defined dually, and a rigid bi-M -simplicial set is simultaneously both a left and a right M -simplicial set, with commuting actions. A bi-M -simplicial set
is the same thing as a left M × M op -simplicial set. Note that it follows from the condition that
M acts by simplicial morphisms that, under the action of M , degenerate n-simplicies are sent
to to degenerate n-simplicies. The geometric realisation construction defines a functor from the
category of left M -simplicial sets (with M -equivariant simplicial morphisms) to the category of
left M -spaces. In particular, this functor associates with each rigid left M -simplicial set a rigid
M -CW complex. Corresponding statements hold for both rigid right and bi-M -simplicial sets.
S
Lemma 4.4. For any rigid left M -simplicial set K = i≥0 Ki the geometric realisation |K| is
a rigid left M -CW complex with respect to the induced action given by
m · [(x, u)/∼] = (m · x, u)/∼.
Proof. It follows from Lemma 4.2 that the action is continuous. By the definition of rigid left
M -simplicial set the M -action maps non-degenerate simplices to non-degenerate simplices, and
the cells of |K| are in natural bijective correspondence with the non-degenerate simplicies of K.
It follows that the action of M on |K| sends n-cells to n-cells. The action is rigid by definition.
Thus |K| is a rigid M -CW complex.

There are obvious right- and bi-M -simplicial set analogues of Lemma 4.4 obtained by replacing M by M op and M × M op , respectively.

18

ROBERT D. GRAY AND BENJAMIN STEINBERG

5. Standard constructions of projective M -CW complexes
In this section we shall give a fundamental method that, for any monoid M , allows us to
construct in a canonical way free left-, right- and bi-M -CW complexes. These constructions
will be important when we go on to discuss M -equivariant classifying spaces later on in the
article. Each of the constructions in this section is a special case of the general notion of the
nerve of a category.
To any (small) category C we can associate a simplicial set N (C) called the nerve of the
category. For each k ≥ 0 we let N (C)k denote the set of all sequences (f1 , . . . , fk ) composable
arrows
f1

f2

f

k
A0 −→ A1 −→ · · · −→
Ak

(5.1)

where we allow objects to repeat in these sequences. The objects of C are the 0-simplices. The
face map di : N (C)k → N (C)k−1 omits Ai , so it carries the above sequence to
f1

f2

fi+1 ◦fi

fi−1

fi+2

f

k
A0 −→ A1 −→ · · · −−−→ Ai−1 −−−−→ Ai+1 −−−→ · · · −→
Ak

while the degeneracy map si : N (C)k → N (C)k+1 carries it to
f1

f2

fi

idA

fi+1

fi+2

f

k
i
A0 −→ A1 −→ · · · −
→ Ai −−→
Ai −
−−
→ Ai+1 −
−−
→ · · · −→
Ak

The classifying space of a (small) category C is the geometric realisation |N (C)| of the nerve
N (C) of C.
The nerve is a functor from Cat (the category of small categories) to SSet (the category
of simplicial sets, with simplicial morphisms) given by applying the functor to the diagram
(5.1). From this it follows that a functor between small categories C and D induces a map of
simplicial sets N (C) → N (D), which in turn induces a continous map between the classifying
spaces |N (C)| → |N (D)|. Also, a natural transformation between two functors between C
and D induces a homotopy between the induced maps on the classifying spaces. In particular,
equivalent categories have homotopy equivalent classifying spaces. Any functor which is left or
right adjoint induces a homotopy equivalence of nerves. Consequently, |N (C)| is contractible if
C admits an initial or final object. (For a proof of this see [Sri96, Corollary 3.7].)
It is obvious from the nerve construction that the nerve of a category which is not connected
is the disjoint union of the nerves of the connected components of the category. Thus, if every
component of C admits an initial or final object, then each of the components of |N (C)| will
be contractible.
It is well know that the geometric realisations of the nerve of a category C and its reversal
C op are homeomorphic.
5.1. The classifying space |BM | of a monoid M . In the general context above, given a
monoid M we can construct a category with a single object, one arrow for every m ∈ M , and
composition given by multiplication. The classifying space |BM | of the monoid M is then the
geometric realisation of the nerve of the category corresponding to M op (the reversal is for
the technical reason of avoiding reversals in the face maps). In more detail, the nerve of this
category is the simplicial set BM with n-simplices: σ = (m1 , m2 , ..., mn ), n-tuples of elements
of M . The face maps are given by


i=0
(m2 , . . . , mn )
di σ = (m1 , . . . , mi−1 , mi mi+1 , mi+2 , . . . , mn ) 0 < i < n


(m1 , . . . , mn−1 )
i = n,

TOPOLOGICAL FINITENESS PROPERTIES

19

and the degeneracy maps are given by
si σ = (m1 , . . . , mi , 1, mi+1 , . . . , mn ) (0 ≤ i ≤ n).
The geometric realisation |BM | is called the classifying space of M . Then |BM | is a CW
complex with one n-cell for every non-degenerate n-simplex of BM , i.e., for every n-tuple all
of whose entries are different from 1. As mentioned in the introduction, classifying spaces of
monoids have received some attention in the literature.
5.2. Right Cayley graph category. Let Γr (M ) denote the right Cayley graph category for
M , which has
• Objects: M ;
• Arrows: (x, m) : x → xm; and
• Composition of arrows: (xm, n) ◦ (x, m) = (x, mn).
The identity at x is (x, 1). This composition underlies our use of M op in defining BM .
−−→
−−→
Let EM be the nerve of the category Γr (M ). The n-simplies of EM may be written using
the notation m(m1 , m2 , ..., mn ) = mτ where τ = (m1 , m2 , ..., mn ) is an n-simplex of BM . Here
m(m1 , m2 , ..., mn ) denotes the n-tuple of composable arrows in the category Γr (M ) where we
start at m and the follow the path labelled by m1 , m2 , ..., mn .
−−→
The face maps in EM are given by


i=0
mm1 (m2 , ..., mn )
di (m(m1 , m2 , ..., mn )) = m(m1 , m2 , ..., mi mi+1 , ..., mn ) 0 < i < n


m(m1 , m2 , ..., mn−1 )
i=n
and the degeneracy maps are given by

si σ = m(m1 , . . . , mi , 1, mi+1 , . . . , mn ) (0 ≤ i ≤ n).
where σ = m(m1 , ..., mn ).
−−→
−−→
−−→
Let |EM | denote the geometric realisation of EM . So |EM | is a CW complex with one
−−→
n-cell for every non-degenerate n-simplex of EM , that is, for every m(m1 , m2 , . . . , mn ) with
−−→
mj 6= 1 for 1 ≤ j ≤ n. As a consequence, by an n-cell of EM we shall mean a non-degenerate
n-simplex.
Consider the right Cayley graph category Γr (M ). For each m ∈ M there is precisely one
morphism (1, m) from 1 to m. Since the category has an initial object we conclude that the
−−→
geometric realisation of its nerve |EM | is contractible.
Applying the nerve functor to the projection functor from the category Γr (M ) to the onepoint category M op , which identifies all the vertices of Γr (M ) to a point, gives a simplicial
−−→
morphism π : EM → BM between the corresponding nerves, which maps
m(m1 , m2 , ..., mn ) 7→ (m1 , m2 , ..., mn ).
−−→
Observe that, for each n, the projection π maps the set of n-cells of EM onto the set of n-cells
−−→
BM . If we then apply the geometric realisation functor we obtain a projection π : |EM | →
|BM | (we abuse notation slightly by using the same notation π to denote this map).
The monoid M acts by left multiplication on the category Γr (M ). By functoriality of the
−−→
nerve, it follows that M acts on the left of EM n by simplicial morphisms via
s · m(m1 , m2 , ..., mn ) = sm(m1 , m2 , ..., mn ).
−−→
Under this action EM n is a free left M -set with basis BMn . Also, if we restrict to the n-cells
(i.e., non-degenerate simplices), then we obtain a free left M -set with basis the set of n-cells of

20

ROBERT D. GRAY AND BENJAMIN STEINBERG

BM . It is an easy consequence of the definitions that this is an action by simplicial morphisms
and that it preserves non-degeneracy in the sense that s · mσ is an n-cell if and only if mσ is an
−−→
−−→
n-cell for all s ∈ M and mσ ∈ EM . Therefore EM is a rigid left M -simplicial set. Combining
−−→
these observations with Lemma 4.4 we conclude that |EM | is a free left M -CW complex which
is contractible.
←−−
Dually, we use EM to denote the nerve of the left Cayley graph category Γl (M ). The
−−→
←−−
simplicial set EM satisfies all the obvious dual statements to those above for EM . In particular
←−−
←−−
M acts freely via right multiplication action on EM by simplicial morphisms, and |EM | is a
free right M -CW complex which is contractible.
←−−→
5.3. Two-sided Cayley graph category. Let Γ(M ) denote the two-sided Cayley graph category for M , which has
• Objects: M × M ;
• Arrows: M × M × M where (mL , m, mR ) : (mL , mmR ) → (mL m, mR ); and
• Composition of arrows: (nL , n, nR ) ◦ (mL , m, mR ) = (mL , mn, nR ) where (mL m, mR ) =
(nL , nnR ). Equivalently this is the same as the composition (mL m, n, nR )◦(mL , m, nnR ) =
(mL , mn, nR ) and corresponds to the path
(mL , mnnR ) → (mL m, nnR ) → (mL mn, nR ).
This is in fact the kernel category of the identity map, in the sense of Rhodes and Tilson [RT89].
←−−→
There is a natural M × M op action of the category Γ(M ).
←−−→
←−→
←−→
Let EM be the nerve of the category Γ(M ). The simplicial set EM parallels the two-sided
←−→
geometric bar construction of J. P. May; see [May72, May75]. The n-simplies of EM may
be written using the notation m(m1 , m2 , ..., mn )s = mτ s where τ = (m1 , m2 , ..., mn ) is an
n-simplex of BM .
←−−→
Here m(m1 , m2 , ..., mn )s denotes the n-tuple of composable morphisms in the category Γ(M )
where we start at (m, m1 m2 . . . mn s) and follow the path
(m, m1 m2 m3 . . . mn s) → (mm1 , m2 m3 . . . mn s) → . . .
. . . (mm1 m2 , m3 . . . mn s) → . . . (mm1 m2 . . . mn , s)
labelled by the edges
(m, m1 , m2 m3 . . . mn s), (mm1 , m2 , m3 . . . mn s), . . . ,
(mm1 m2 . . . mn−2 , mn−1 , mn s), (mm1 m2 . . . mn−1 , mn , s)
←−→
and finish at (mm1 m2 . . . mn , s). The face maps in the nerve EM are given by


i=0
mm1 (m2 , ..., mn )s
di (m(m1 , m2 , ..., mn )s) = m(m1 , m2 , ..., mi mi+1 , ..., mn )s 0 < i < n


m(m1 , m2 , ..., mn−1 )mn s
i=n
and the degeneracy maps are given by

si σ = m(m1 , . . . , mi , 1, mi+1 , . . . , mn )s (0 ≤ i ≤ n),
where σ = m(m1 , ..., mn )s.
←−→
←−→
←−→
Let |EM | denote the geometric realisation of EM . So |EM | is a CW complex with one n-cell
←−→
for every non-degenerate n-simplex of EM . Observe that (mL , mR ) and (m′L , m′R ) are in the
←−−→
same component of the two-sided Cayley graph category Γ(M ) if and only if mL mR = m′L m′R .
Moreover, for each m ∈ M the vertex (1, m) is initial in its component. It follows from

TOPOLOGICAL FINITENESS PROPERTIES

21

←−→
←−→
these observations that π0 (|EM |) ∼
= M as an M × M op -set, and each component of |EM | is
←−−→
contractible. There is a natural projection from Γ(M ) to the one-point category M op mapping
(mL , m, mR ) to its middle component m. Applying the nerve functor to this projection gives a
←−→
simplicial morphism π : EM → BM given by
m(m1 , m2 , ..., mn )s 7→ (m1 , m2 , ..., mn ).
←−→
As in the one-sided case, this projection sends n-cells to n-cells and induces a map π : |EM | →
|BM | between the corresponding geometric realisations.
←−→
The monoid M has a natural two-sided action on EM n via
x · [m(m1 , m2 , ..., mn )s] · y = xm(m1 , m2 , ..., mn )sy.
←−→
Under this action EM is a free rigid bi-M -simplicial set. Combining these observations with
←−→
←−→
Lemma 4.4 we conclude that |EM | is a free bi-M -CW complex such that π0 (|EM |) ∼
= M as an
←−→
op
M × M -set and each component of |EM | is contractible
6. One-sided classifying spaces and finiteness properties
We will define left and right equivariant classifying spaces for a monoid M . Two-sided
equivariant classifying spaces will be defined in the next section. As we shall see, the examples
discussed in Section 5 will serve as the standard models of such spaces.
We say that a projective M -CW complex X is a (left) equivariant classifying space for M if it
is contractible. A right equivariant classifying space for M will be a left equivariant classifying
space for M op . Notice that the augmented cellular chain complex of an equivariant classifying
space for M provides a projective resolution of the trivial (left) ZM -module Z.
Example 6.1. The bicyclic monoid is the monoid B with presentation ha, b | ab = 1i. It is not
hard to see that each element of B is uniquely represented by a word of the form bi aj where
i, j ≥ 0. Figure 1 shows an equivariant classifying space for B. The 1-skeleton is the Cayley
graph of B and there is a 2-cell glued in for each path labelled ab. This example is a special
case of far more general results about equivariant classifying spaces of special monoids which
will appear in a future paper of the current authors [GS].
..
.

..
.

a

b

a

b

a

b

a

b

b

b

···

Figure 1. An equivariant classifying space for the bicyclic monoid
Our first goal is to show that any two equivariant classifying spaces for M are M -homotopy
equivalent.
Lemma 6.2. Let X be an equivariant classifying space for M and let Y be a contractible
M -space. Then there exists a continuous M -equvariant mapping f : X → Y .

22

ROBERT D. GRAY AND BENJAMIN STEINBERG

Proof. The proof constructs by induction M -equivariant
` continuous mappings fn : Xn → Y with
fn extending fn−1 . To define f0 , observe Q
that X0 = a∈A M ea (without loss of generality) and
so, by Proposition 2.3, TopM (X0 , Y ) ∼
= a∈A ea Y 6= ∅ and so we can define f0 . Assume that
fn : Xn → Y has been defined. Let Z be the one-point space with the trivial M -action and let
k : Y → Z be the unique M -equivariant map. Then k is a weak equivalence. So by Theorem 2.5
we can construct a commutative diagram
Xn

i0

Xn × I

i1

i

Z

k

Y

Xn
fn
i
fn+1

Xn+1

i0

Xn+1 × I

i1

Xn+1

with fn+1 M -equivariant. Now take f to be the colimit of the fn .



Theorem 6.3. Let X, Y be equivariant classifying spaces for M . Then X and Y are M homotopy equivalent by a cellular M -homotopy equivalence.
Proof. By Corollary 2.7 and Theorem 2.8 it suffices to construct an M -equivariant continuous
mapping f : X → Y . But Lemma 6.2 does just that.

Next we now give an elementary proof that contractible free M -CW complexes exist and
hence there are equivariant classifying spaces for M . A more canonical construction, using
simplicial sets, was given in the previous section.
Lemma 6.4. Let M be a monoid.
(1) If X0 is a non-empty projective (free) M -set, then there is a connected projective (free)
M -graph X with vertex set X0 .
(2) If X is a connected projective (free) M -CW complex such that πq (X) = 0 for 0 ≤ q < n,
then there exists a projective M -CW complex Y containing X as a projective M -CW
subcomplex and such that Yn = Xn and πq (Y ) = 0 for 0 ≤ q ≤ n.
(3) If X is a connected projective (free) M -CW complex such that πq (X) is trivial for
0 ≤ q < n, then there exists a contractible projective (free) M -CW complex Y containing
X as a projective M -CW subcomplex and such that Yn = Xn .
Proof. For the first item, fix x0 ∈ X0 . The edge set of X will be bijection with M × X0 with the
edge corresponding to (m, x) connecting mx0 to mx. Then X is a projective (free) M -graph
and each vertex x is connected to x0 via the edge (1, x).
For the second item, we show that we can add free M -cells of dimension n + 1 to X to obtain
a new projective M -CW complex Y with πn (Y ) = 0. If πn (X) = 0, then take Y = X. So
assume that πn (X) is non-trivial. Fix a base point x0 ∈ X0 and let fa : S n → X, a ∈ A, be
mappings whose based homotopy classes form a set of generators for πn (X, x0 ). As Xn → X is
an n-equivalence, we may assume without loss of generality that fa : S n → Xn . Suppose that
X is constructed from pushouts as per (2.1). Note that M × A, where A has the trivial action,
is a free M -set. Let us define Y by putting Yk = Xk for 0 ≤ k ≤ n, defining Yn+1 to be the
pushout
`
(Pn+1 × S n ) (M × A × S n )
Xn
(Pn+1 × B n+1 )

`

(M × A × B n+1 )

Yn+1 ,

TOPOLOGICAL FINITENESS PROPERTIES

23

where the top map is the union of the attaching map for X with the mapping (m, a, x) 7→ mfa (x)
(cf. Proposition 2.3), and putting Yk = Xk ∪ Yn+1 for k > n + 1. Then Y is a projective M -CW
complex containing X as a projective M -CW subcomplex and with Yn = Xn . Moreover, since
Xn = Yn → Y is an n-equivalence, it follows that the based homotopy classes of the fa : S n → Y
generate πn (Y, x0 ). By construction the fa can be extended to B n+1 → Y and so their classes
are trivial in πn (Y, x0 ). Thus πn (Y ) = 0. Also, because Xn = Yn → Y is an n-equivalence, we
have that πq (Y ) = 0 for 0 ≤ q < n.
The final item follows from Whitehead’s theorem, iteration of the second item and that
Yn → Y is an n-equivalence for all n ≥ 0.

Corollary 6.5. Let M be a monoid. Then there exists a contractible free M -CW complex.
Proof. Put X0 = M . Then by Lemma 6.4 we can find a connected free M -graph X with vertex
set X0 . Now applying the third item of Lemma 6.4 we can find a contractible free M -CW
complex with 1-skeleton X.

Example 6.6. It follows from the definitions and results in Section 5 that the geometric realisa−−→
tion |EM | of the nerve of the right Cayley graph category of M is a left equivariant classifying
space for M .
Corollary 6.7. If X, Y are equivariant classifying spaces for M , then M \X and M \Y are
homotopy equivalent. In particular, M \X ≃ |BM |. Therefore, if G is a group and X is an
equivariant classifying space for G, then G\X is an Eilenberg-Mac Lane space for G. Conversely, the universal cover of any Eilenberg-Mac Lane space for G is an equivariant classifying
space for G.
Proof. The first statement follows from Theorem 6.3 and Proposition 3.5. The second statement
−−→
follows from the first as |EM | is an equivariant classifying space for M . The group statements
then follow from the previous statements and classical covering space theory.

If M and N are monoids, then E(M ×N ) = E(M )×E(N ) and (M ×N )(e, f ) = M e×N f . It
follows that if P is a (finitely generated) projective M -set and Q a (finitely generated) projective
N -set, then P × Q is a (finitely generated projective) M × N -set.
Proposition 6.8. Let M, N be monoids and let X, Y be equivariant classifying spaces for M, N ,
respectively. Then X × Y is an M × N -equivariant classifying space, which is of M × N -finite
type whenever X is of M -finite type and Y is of N -finite type.
Proof. Assume that X is obtained via attaching projective M -cells Pn × B n and that Y is
obtained by
N -cells Qn × B n . Then the n-cells of X × Y are obtained by
`nattaching projective
adjoining i=0 Pi × Qn−i × B n and hence X × Y is an M × N -projective CW complex which
is of M × N -finite type whenever X is of M -finite type and Y is of N -finite type. As X × Y is
contractible, we deduce that it is an M × N -equivariant classifying space.

6.1. Monoids of type left-Fn . A key definition for this paper is the following. A monoid M
is of type left-Fn if there is an equivariant classifying space X for M such that Xn is M -finite,
i.e., such that M \X has finite n-skeleton. We say that M is of type left-F∞ if M has an
equivariant classifying space X that is of M -finite type, i.e., M \X is of finite type. The monoid
M is defined to have type right-Fn if M op is of type left-Fn for 0 ≤ n ≤ ∞. The following
proposition contains some basic facts.
Proposition 6.9. Let M be a monoid.
(1) A group is of type left-Fn if and only if it is of type Fn in the usual sense for any
0 ≤ n ≤ ∞.

24

ROBERT D. GRAY AND BENJAMIN STEINBERG

(2) For 0 ≤ n ≤ ∞, if M is of type left-Fn , then it is of type left-FPn .
(3) If M is of type left-F∞ , then it is of type left-Fn for all n ≥ 0.
Proof. The first item follows from Corollary 6.7 and Corollary 3.3. The second is immediate
using that the augmented cellular chain complex of an equivariant classifying space X gives a
projective ZM -resolution of the trivial ZM -module since if X is built up from pushouts as per
(2.1), then the nth -chain module is isomorphic to ZPn . The final item is trivial.

−−→
Note that, trivially, if M is a finite monoid then |EM | has finitely many cells in each dimension
and thus M is of type left-F∞ .
Sometimes it will be convenient to use the following reformulation of the property left-Fn .
Proposition 6.10. Let M be a monoid. The following are equivalent for 0 ≤ n < ∞.
(1) M is of type left-Fn
(2) There is a connected M -finite projective M -CW complex X of dimension at most n with
πq (X) = 0 for 0 ≤ q < n.
Proof. If Y is an equivariant classifying space for M such that Yn is M -finite, then since Yn → Y
is an n-equivalence, we deduce that X = Yn is as required for the second item. Conversely, if
X is as in the second item, we can construct by Lemma 6.4 an equivariant classifying space Y
for M with Yn = X. Thus M is of type left-Fn .

Recall that the fundamental group of |BM | is isomorphic to the universal group (or maximal
group image, or group completion) U (M ) of M i.e., the group with generators M and relations
the multiplication table of M (cf. [GZ67]).
Corollary 6.11. Let M be a monoid. If M is of type left-F1 , then U (M ) is finitely generated.
If M is of type left-F2 , then U (M ) is finitely presented.
Proof. By Corollary 6.7, |BM | in the first case is homotopy equivalent to a CW complex with
finite 1-skeleton and in the second case to a CW complex with finite 2-skeleton by Corollary 3.3.
Thus U (M ) ∼

= π1 (BM ) has the desired properties in both cases.
Recall that an inverse monoid is a monoid M with the property that for every m ∈ M there
is a unique element m′ ∈ M such that mm′ m = m and m′ mm′ = m′ . For more on inverse
monoids, and other basic concepts from semigroup theory we refer the reader to [How95].
Corollary 6.12. Let M be a monoid such that |BM | is an Eilenberg-Mac Lane space (e.g., if
M is cancellative with a left or right Ore condition or if M is an inverse monoid) and suppose
that 0 ≤ n ≤ ∞. If M is of type left-Fn , then U (M ) is of type Fn .
Proof. If X is an equivariant classifying space for M , then M \X is homotopy equivalent to
|BM | by Corollary 6.7 and hence is an Eilenberg-Mac Lane space for U (M ). The result now
follows from Corollary 3.3.

Since, as already mentioned above, D. McDuff [McD79] has shown that every path-connected
space has the weak homotopy type of the classifying space of some monoid, not every monoid
|BM | is an Eilenberg-Mac Lane space. So not every monoid satisfies the hypotheses of Corollary 6.12. The fact that if M is cancellative with a left or right Ore condition then |BM | is
an Eilenberg-Mac Lane space is well known. If M is an inverse monoid then |BM | may also
be shown to be an Eilenberg-Mac Lane space. Both of these results can easily be proved by
appealing to Quillen’s theorem A, see [Wei13, Chapter 4], and should be considered folklore.

TOPOLOGICAL FINITENESS PROPERTIES

25

The converse of Corollary 6.12 does not hold. For example, the free inverse monoid on one
generator is not of type left-F2 while its maximal group image Z is F∞ (this proof of the fact
that the free inverse monoid on one generator is not left-F2 will appear in [GS]).
For groups, being of type F1 is equivalent to finite generation. For monoids, the condition of
being left-F1 is considerably weaker. Recall that if M is a monoid and A ⊆ M , then the (right)
Cayley digraph Γ(M, A) of M with respect to A is the graph with vertex set M and with edges
in bijection with M × A where the directed edge (arc) corresponding to (m, a) starts at m and
ends at ma. Notice that Γ(M, A) is a free M -CW graph and is M -finite if and only if A is
finite.
Theorem 6.13. Let M be a monoid. The following are equivalent.
(1) M is of type left-F1 .
(2) M is of type left-FP1 .
(3) There is a finite subset A ⊆ M such that Γ(M, A) is connected as an undirected graph.
In particular, any finitely generated monoid is of type left-F1 .
Proof. Item (1) implies (2) by Proposition 6.9, whereas (2) implies (3) by a result due to
Kobayashi [Kob07]. For completeness, let us sketch the proof that (2) implies (3). Let ε : ZM →
Z be the augmentation map; the ideal I = ker ε is called the augmentation ideal. If M is of
type left-FP1 , then I must be finitely generated because the augmentation map gives a partial
free resolution. But I is generated by all elements of the form m − 1 with m ∈ M . Hence there
is a finite subset A ⊆ M such that the elements a − 1 with a ∈ A generate I. Consider the
Cayley digraph Γ(M, A). Then M acts cellularly on Γ(M, A) and hence acts on π0 (Γ(M, A)).
There is a surjective ZM -module homomorphism η : ZM → Zπ0 (Γ(M, A)) mapping m ∈ M to
the connected component of the vertex m of Γ(M, A). Moreover, the augmentation ε factors
through η. Thus to show that Γ(M, A) is connected, it suffices to show that I = ker η. By
construction ker η ⊆ I. But if a ∈ A, then a and 1 are in the same connected component of
Γ(M, A) and thus a − 1 ∈ ker η. Since the a − 1 with a ∈ A generate I, we deduce that I ⊆ ker η
and hence Γ(M, A) is connected.
Finally, (3) implies (1) by Proposition 6.10 as Γ(M, A) is an M -finite connected free M -CW
complex of dimension at most 1.

We next show that a finitely presented monoid is of type left-F2 . In fact, we shall see later
that finitely presented monoids are of type bi-F2 , which implies left-F2 , but the proof of this
case is instructive.
Theorem 6.14. Let M be a finitely presented monoid. Then M is of type left-F2
Proof. Suppose that M is generated by a finite set A with defining relations u1 = v1 , . . . , un =
vn . Let us construct a 2-dimensional, M -finite, free M -CW complex X with 1-skeleton the
Cayley graph Γ(M, A) by attaching a free M -cell M × B 2 for each relation. Let pi , qi be the
paths from 1 to mi labelled by ui and vi , respectively, where mi is the image of ui (and vi ) in M .
Then we glue in a disk di with boundary path pi qi−1 and glue in M ×B 2 using Proposition 2.3 (so
{m} × B 2 is sent to mdi ). Then X is an M -finite connected free M -CW complex of dimension
at most 2. See Figure 1 for this construction for the bicyclic monoid. By Proposition 6.10, it
suffices to prove that X is simply connected.
A digraph is said to be rooted if there is a vertex v so that there is a directed path from v to
any other vertex. For instance, Γ(M, A) is rooted at 1. It is well known that a rooted digraph
admits a spanning tree, called a directed spanning tree, such that the geodesic from the root to
any vertex is directed. Let T be a directed spanning tree for Γ(M, A) rooted at 1 (this is the
same thing as a prefix-closed set of normal forms for M so, for instance, shortlex normal forms

26

ROBERT D. GRAY AND BENJAMIN STEINBERG
a

would do). Let e = m −→ ma be a directed edge not belonging to T . Then the corresponding
generator of π1 (X, 1) is of the form peq −1 where p and q are directed paths from 1 to m and
ma, respectively. Let u be the label of p and v be the label of q. Then ua = v in M . Thus it
suffices to prove that if x, y ∈ A∗ are words which are equal in M to an element m′ , then the
loop ℓ labelled xy −1 at 1, corresponding to the pair of parallel paths 1 to m′ labelled by x and
y, is null homotopic. By induction on the length of a derivation from x to y, we may assume
that x = wui w′ and y = wvi w′ for some i = 1, . . . , n. Let m0 be the image of w in M . Then
m0 di is a 2-cell with boundary path the loop at m0 labeled by ui vi−1 . It follows that ℓ is null
homotopic. This completes the proof.

The converse of Theorem 6.14 is not true, e.g., the monoid (R, ·) is of type left-F2 (by
Corollary 6.23) but is not even finitely generated. It is natural to ask whether there is a nice
characterisation, analogous to Theorem 6.13(3), for left-F2 in terms of the right Cayley graph
together with the left action of M . We would guess that M is of type left-F2 if and only if it
has a finite subset A ⊆ M such that Γ(M, A) is connected, and finitely many free M -2-cells
can be adjoined to make a simply connected 2-complex.
It is well known that, for finitely presented groups, the properties Fn and FPn are equivalent
for 3 ≤ n ≤ ∞. We now provide the analogue in our context. Here we replace finitely presented
by left-F2 .
Theorem 6.15. Let M be a monoid of type left-F2 . Then M is of type left-Fn if and only if
M is of type left-FPn for 0 ≤ n ≤ ∞.
Proof. We prove that if there is a connected M -finite projective M -CW complex X of dimension
at most n with πq (X) = 0 for 0 ≤ q < n with n ≥ 2 and M is of type left-FPn+1 , then there
is a connected M -finite projective M -CW complex Y of dimension at most n + 1 with Yn = X
and πq (Y ) = 0 for all 0 ≤ q < n + 1. This will imply the theorem by Proposition 6.10,
Proposition 6.9 and induction.
Since X is simply connected, Hq (X) = 0 for 1 ≤ q < n and Hn (X) ∼
= πn (X) by the Hurewicz
theorem. Therefore, the augmented cellular chain complex of X gives a partial projective
resolution of Z of length at most n, which is finitely generated in each degree. Therefore,
since M is of type left-FPn+1 , it follows that Hn (X) = ker dn : Cn (X) → Cn−1 (X) is finitely
generated as a left ZM -module. Choose representatives of fa : S n → X, with a ∈ A, of a
finite set of elements of πn (X) that map to a finite ZM -module generating set of Hn (X) under
the Hurewicz isomorphism. Then form Y by adjoining M × A × B n+1 to X via the attaching
map M × A × S n → Xn given by (m, a, x) 7→ mfa (x). Then Y is an M -finite projective
M -CW complex of dimension n + 1 with Yn = X. Since the inclusion of X = Yn into Y is
an n-equivalence, we deduce that πq (Y ) = 0 for 1 ≤ q < n and that the inclusion X → Y is
surjective on πn . But since the Hurewicz map in degree n is natural and is an isomorphism for
both X and Y , we deduce that the inclusion X → Y induces a surjection Hn (X) → Hn (Y ).
But, by construction, the images of the ZM -module generators of Hn (X) are trivial in Hn (Y )
(since they represent trivial elements of πn (Y )). We deduce that Hn (Y ) = 0 and hence, by the
Hurewicz theorem, πn (Y ) = 0. This completes the induction.

Notice that Theorem 6.15 implies that M is of type left-F∞ if and only if M is of type left-Fn
for all n ≥ 0.
Proposition 6.16. If M is of type left-Fn with n ≥ 1, then M has a free contractible M -CW
complex X such that Xn is M -finite.
Proof. This is clearly true for n = 1 by Theorem 6.13. Note that Lemma 6.4 and the construction in the proof of Theorem 6.15 show that if Y is a simply connected M -finite free M -CW

TOPOLOGICAL FINITENESS PROPERTIES

27

complex of dimension at most 2 and M is of type left-FPn , then one can build a contractible
free M -CW complex X with X2 = Y such that Xn is M -finite. Thus it remains to prove that
if there is a simply connected M -finite projective M -CW complex Y of dimension 2, then there
is a simply connected
` M -finite free M -CW complex X of dimension 2.
Note that Y0 = a∈A M ea with A a finite set. Define X0 = M × A. Identifying M ea with
M ea × {a}, we may view Y0 as an M -subset of X0 . Using this identification, we can define X1
to consists of Y1 (the edges of Y ) along with some new edges. We glue in an edge from (m, a)
to (mea , a) for each m ∈ M and a ∈ A; that is we glue in a free M -cell M × A × B 1 where
the attaching map takes (m, a, 0) to (m, a) and (m, a, 1) to (mea , a). Notice that all vertices of
X0 \ Y0 are connected to a vertex of Y0 in X1 and so X1 is connected as Y1 was connected.
To define X2 , first we keep all the two-cells from Y2 . Notice that if T is a spanning tree for Y ,
then a spanning tree T ′ for X can be obtained by adding to T all the edges (m, a) −→ (mea , a)
with m ∈
/ M ea (all vertices of X0 \ Y0 have degree one). Thus the only edges in X1 \ Y1 that do
not belong to T ′ are the loop edges (m, a) −→ (mea , a) for m ∈ M ea that we have added. So if
we attach M × A × B 2 to X1 by the attaching map M × A × S 1 → X1 mapping {m} × {a} × S 1
to the loop edge (mea , a) −→ (mea , a) from X1 \ Y1 , then we obtain a simply connected free
M -CW complex X which is M -finite. This completes the proof.

In light of Proposition 6.16, one might wonder why we bother allowing projective M -CW
complexes rather than just free ones. The reason is because projective M -CW complexes are
often easier to construct and, as we are about to see, sometimes it is possible to find an M finite equivariant classifying space for M which is projective when no M -finite free equivariant
classifying space exists. This will be relevant when considering geometric dimension.
Let M be a non-trivial monoid with a right zero element z. Then M z = {z} is a one-element
set with the trivial action. Since z is idempotent, it follows that the trivial M -set is projective
but not free. Therefore, the one-point space with the trivial M -action is an M -finite equivariant
classifying space for M , which is not free. We will show that if M has a finite number of right
zeroes (e.g., if M has a zero element), then there is no finite free resolution of the trivial module
which is finitely generated in each degree. In this case, every free equivariant classifying space
for M of M -finite type will be infinite dimensional.
A finitely generated projective module P over a ring R is said to be stably free if there are
finite rank free modules F, F ′ such that P ⊕ F ′ ∼
= F . The following lemma is well known, but
we include a proof for completeness.
Lemma 6.17. Let P be a finitely generated projective (left) module over a ring R. Then P has
a finite free resolution, finitely generated in each degree, if and only if P is stably free.
Proof. Suppose that P is stably free, say P ⊕ F ′ ∼
= F with F, F ′ finite rank free R-modules.
Then the exact sequence
0 −→ F ′ −→ F −→ P −→ 0
provides a finite free resolution of P that is finitely generated in each degree.
Conversely, suppose that
0 −→ Fn −→ Fn−1 −→ · · · −→ F0 −→ P −→ 0
is a free resolution with Fi finitely generated for all 0 ≤ i ≤ n. We also have a projective
resolution
0 −→ Pn −→ Pn−1 −→ · · · −→ P0 −→ P −→ 0
with P0 = P and Pi = 0 for 1 ≤ i ≤ n because P is projective. By the generalized Schanuel’s
lemma, we have that
P0 ⊕ F1 ⊕ P2 ⊕ F3 ⊕ · · · ∼
= F0 ⊕ P1 ⊕ F2 ⊕ P3 ⊕ · · ·

28

ROBERT D. GRAY AND BENJAMIN STEINBERG

and hence
P ⊕ F1 ⊕ F3 ⊕ · · · ∼
= F0 ⊕ F2 ⊕ · · ·
and so P is stably free.



So we are interested in showing that the trivial module for ZM is not stably free if M is a
non-trivial monoid with finitely many right zeroes (and at least one).
Recall that a ring R is said to have the Invariant Basis Number property (IBN) if whenever
Rn ∼
= Rm as R-modules, one has m = n (where m, n are integers); in this definition it does not
matter if one uses left or right modules [Lam99].
Our first goal is to show that if M is a monoid with zero z, then the contracted monoid
ring ZM/Zz has IBN. This result is due to Pace Nielsen, whom we thank for allowing us to
reproduce it. It is equivalent to show that if M is a monoid and I is a proper ideal of M , then
ZM/ZI has IBN. The proof makes use of the Hattori-Stallings trace (see [Wei13, Chapter 2]).
Let ∼ be the least equivalence relation on a monoid M such that mn ∼ nm for all m, n ∈ M ;
this relation, often called conjugacy, has been studied by a number of authors.
Lemma 6.18. Let M be a monoid and e ∈ M an idempotent. Suppose that e is conjugate to
an element of an ideal I. Then e ∈ I.
Proof. Suppose that e is conjugate to m ∈ I. Then we can find elements x1 , . . . , xn , y1 , . . . , yn ∈
M with e = x1 y1 , yi xi = xi+1 yi+1 and yn xn = m. Then e = en+1 = (x1 y1 )n+1 = x1 (y1 x1 )n y1 =
x1 (x2 y2 )n y1 = x1 x2 (y2 x2 )n−1 y2 y1 = · · · = x1 · · · xn (yn xn )yn yn−1 · · · y1 ∈ I as yn xn = m ∈
I.

If R is a ring, then [R, R] denotes the additive subgroup generated by all commutators
ab − ba with a, b ∈ R. The abelian group R/[R, R] is called the Hattori-Stallings trace of R;
this is also the 0-Hochschild homology group of R. Cohn proved that if 1 + [R, R] has infinite
order in R/[R, R], then R has IBN (see [Lam99, Exercise 1.5]). The point is that if A is an
m × n matrix over R and B is an n × m-matrix over R with AB = Im and BA = In , then
m+[R, R] = T (AB) = T (BA) = n+[R, R] where, for a square matrix C over R, we define T (C)
to be the class of the sum of the diagonal entries of C in R/[R, R]. The following proposition
is an elaboration of Pace Nielsen’s argument (see [Nie]).
Proposition 6.19. Let M be a monoid and I a (possibly empty) ideal. Let R = ZM/ZI.
Then R/[R, R] is a free abelian group on the conjugacy classes of M that do not intersect I.
More precisely, if T is a transversal to the conjugacy classes of M not intersecting I, then the
elements of the form m + [R, R] with m ∈ T form a basis for R/[R, R].
Proof. We view R as having basis M \ I subject to the relations of the multiplication table
of M and that the elements of I are 0. Let A be the free abelian group on the conjugacy
classes of M that do not intersect I. Write [m] for the conjugacy class of m ∈ M . Define
an abelian group homomorphism f : A → R/[R, R] by f ([m]) = m + [R, R]. This is well
defined because xy + [R, R] = yx + [R, R] for x, y ∈ M with xy, yx ∈
/ I. To see that f
is surjective, note that if m ∈ M \ I with [m] ∩ I 6= ∅, then m + [R, R] = [R, R]. This
follows because if m = x1 y1 , yi xi = xi+1 yi+1 , for i = 1, . . . , n − 1, and yn xn ∈ I, then
[R, R] = yn xn + [R, R] = xn yn + [R, R] = · · · = y1 x1 + [R, R] = m + [R, R].
Let us define g : R → A on m ∈ M \ I by
(
[m], if [m] ∩ I = ∅
g(m) =
0,
else.

TOPOLOGICAL FINITENESS PROPERTIES

Then if a, b ∈ R with, say,
a=

X

cm m, b =

m∈M \I

then we have that

ab − ba =

X

29

dn n

n∈M \I

X

cm dn (mn − nm).

m,n∈M \I

Since mn ∼ nm, either both map to 0 under g or both map to [mn] = [nm]. Therefore,
ab − ba ∈ ker g and so g induces a homomorphism g′ : R/[R, R] → A. Clearly, if [m] ∩ I = ∅,
then gf ([m]) = g′ (m + [R, R]) = g(m) = [m]. It follows that f is injective and hence an
isomorphism. The result follows.

As a consequence we deduce the result of Nielsen.
Corollary 6.20. Let M be a monoid and I a proper ideal (possibly empty). Then ZM/ZI has
IBN. In particular, contracted monoid rings have IBN.
Proof. Put R = ZM/ZI. If I is a proper ideal, then 1 is not conjugate to any element of I by
Lemma 6.18. It follows from Proposition 6.19 that 1 + [R, R] has infinite order in R/[R, R] and
hence R has IBN.

Theorem 6.21. Let M be a non-trivial monoid with a finitely many right zeroes (and at least
one). Then the trivial left ZM -module Z is projective but not stably free and hence does not
have a finite free resolution that is finitely generated in each degree.
Proof. Let I be the set of right zero elements of M and fix z ∈ I. Observe that I is a proper
two-sided ideal. Note that z, 1 − z form a complete set of orthogonal idempotents of ZM and
so ZM ∼
= ZM z ⊕ ZM (1 − z) and hence ZM z ∼
= Z is projective. Suppose that Z is stably free,
′ free ZM -modules of rank r, r ′ , respectively.
F
with
F,
F
that is, Z ⊕ F ′ ∼
=
There is a exact functor from ZM -modules to zZM z-modules given by V 7→ zV . Note that
zZM z = Zz ∼
= Z as a ring. Also, zZM = ZI is a free abelian group (equals zZM z-module) of
rank |I| < ∞. Therefore,
′
′
Z|I|r ∼
= (ZI)r ∼
= zF ∼
= Z ⊕ zF ′ ∼
= Z ⊕ (ZI)r ∼
= Z1+|I|r

as Z-modules and hence r|I| = 1 + r ′ |I| as Z has IBN. But putting R = ZM/ZI and observing
that Z/ZI · Z = 0, we have that
′
Rr ∼
= R ⊗ZM F ∼
= (R ⊗ZM Z) ⊕ (R ⊗ZM F ′ ) ∼
= Rr

and hence r = r ′ as R has IBN by Corollary 6.20. This contradiction completes the proof.



There is, of course, a dual result for left zeroes. In particular, if M is non-trivial monoid with
a zero element, then Z is not stably free as either a left or right ZM -module. Thus, if M is a
non-trivial monoid with zero, it has no M -finite free left or right equivariant classifying space
but it has an M -finite projective one. This justifies considering projective M -CW complexes.
If L is a left ideal of M containing an identity e, then L = M e = eM e. Note that ϕ : M → M e
given by ϕ(m) = me is a surjective monoid homomorphism in this case since ϕ(1) = e and
ϕ(mn) = mne = mene = ϕ(m)ϕ(n) as ne ∈ M e = eM e. Also note that the left M -set structure
on M e given by inflation along ϕ corresponds to the left M -set structure on M e induced by left
multiplication because if m ∈ M and n ∈ M e, then n = en and so mn = men = ϕ(m)n. Notice
that if f ∈ E(M e), then f ∈ E(M ) and M ef = M f is projective as both an M e-set and an M set. Thus each (finitely generated) projective M e-set is a (finitely generated) projective M -set
via inflation along ϕ. Note that if A is a left M -set, then eM ⊗M A ∼
= eA via em ⊗ a 7→ ema.

30

ROBERT D. GRAY AND BENJAMIN STEINBERG

Proposition 6.22. Suppose that M and e ∈ E(M ) with M e = eM e and 0 ≤ n ≤ ∞. If M e
is of type left-Fn , then so is M . The converse holds if eM is a finitely generated projective left
eM e-set.
Proof. If X is a projective M e-CW complex constructed via pushouts as in (2.1) (but with
M replaced by M e), then each Pn is a projective M -set via inflation along ϕ and so X is a
projective M -CW complex. Moreover, if X is of M e-finite type (respectively, M e-finite), then
it is of M -finite type (respectively, M -finite). Thus if M e is of type left-Fn , then so is M .
Suppose that X is an equivariant classifying space for M and eM is a finitely generated
projective left eM e-set. Then eM ⊗M X ∼
= eX. Now eX is a projective eM e-CW complex and
if Xn is M -finite, then (eX)n = eXn is eM e-finite by Corollary 3.2. Moreover, since eX is a
retract of X as a CW complex and X is contractible, it follows that eX is contractible. Thus
eX is an equivariant classifying space for eM e. The result follows.

Our first corollary is that having a right zero guarantees the property left-F∞ (which can be
viewed as a defect of the one-sided theory).
Corollary 6.23. If M contains a right zero, then M is of type left-F∞ . Hence any monoid
with a zero is both of type left- and right-F∞ .
Proof. If e is a right zero, then M e = {e} = eM e and {e} is of type left-F∞ . Thus M is of type
left-F∞ by Proposition 6.22.

Recall that two elements m and n a monoid M are said to be L -related if and only if they
generate the same principal left ideal, i.e., if M m = M n. Clearly L is an equivalence relation
on M .
Corollary 6.24. Suppose that M is a monoid and e ∈ E(M ) with eM a two-sided minimal
ideal of M and 0 ≤ n ≤ ∞. Note that Ge = eM e is the maximal subgroup at e. If Ge is of
type-Fn , then M is of type left-Fn . If eM contains finitely many L -classes, then the converse
holds.
Proof. Note that M e = eM e = Ge and so the first statement is immediate from Proposition 6.22. For the converse, it follows from Green’s lemma [How95, Lemma 2.2.1] that eM is
free left Ge -set and that the orbits are the L -classes of eM . Thus the second statement follows
from Proposition 6.22.

Corollary 6.24 implies that if M is a monoid with a minimal ideal that is a group G, then
M is of type left-Fn if and only if G is of type Fn and dually for right-Fn .
The following is a slight extension of the fact that a finite index subgroup of a group of type
Fn is also of type Fn ; see [Bro94, Chapter VIII, Proposition 5.1].
Proposition 6.25. Let M be a monoid and N a submonoid such that M is a finitely generated
projective left N -set. If M is of type left-Fn , then N is of type left-Fn , as well.
Proof. Observe that each finitely generated free left M -set is a finitely generated projective
N -set. Hence each finitely generated projective M -set, being a retract of a finitely generated
free left M -set, is a finitely generated projective N -set. Thus any equivariant classifying space
for M is also an equivariant classifying space for N .

An immediate consequence of Proposition 6.8 is the following.
Proposition 6.26. Let M, N be monoids of type left-Fn . Then M × N is of type left-Fn .

TOPOLOGICAL FINITENESS PROPERTIES

31

6.2. Left geometric dimension. Let us define the left geometric dimension of M to be
the minimum dimension of a left equivariant classifying space for M . The right geometric
dimension is, of course, defined dually. Clearly, the geometric dimension is an upper bound on
the cohomological dimension cd M of M . Recall that the (left) cohomological dimension of M
is the projective dimension of the trivial module Z, that is, the shortest length of a projective
resolution of Z. As mentioned in the introduction, for groups of cohomological dimension
different than 2, it is known that geometric dimension coincides with cohomological dimension,
but the general case is open.
Theorem 6.27. Let M be a monoid. Then M has an equivariant classifying space of dimension
max{cd M, 3}.
Proof. If M has infinite cohomological dimension, then this is just the assertion that M has an
equivariant classiyfing space. So assume that cd M < ∞. Put n = max{cd M, 3}. Let Y be
an equivariant classifying space for M . As the inclusion Yn−1 → Y is an (n − 1)-equivalence,
we deduce that πq (Yn−1 ) is trivial for 0 ≤ q < n − 1. Also, as the augmented cellular chain
complex of Yn−1 provides a partial projective resolution of the trivial module of length n−1 and
cd M ≤ n, it follows that ker dn−1 = Hn−1 (Yn−1 ) is a projective ZM -module. By the Eilenberg
swindle, there is a free ZM -module F such that Hn−1 (Yn−1 ) ⊕ F ∼
= F . Suppose that F is free
on a set A. Fix a basepoint y0 ∈ Yn−1 . We glue a wedge of (n − 1)-spheres, in bijection with
A, into Yn−1 at y0 as well as freely gluing in its translates. That is we form a new projective
M -CW complex Z with Zn−2 = Yn−2 and where Z = Zn−1 consists of the (n − 1)-cells from
Yn−1 and M × A × B n−1 where the attaching map M × A × S n−2 is given by (m, a, x) 7→ my0 .
Notice that Cn−1 (Z) ∼
= Cn−1 (Yn−1 ) ⊕ F as a ZM -module and that the boundary map is zero
on the F -summand since the boundary of each of the new (n − 1)-cells that we have glued in is
a point and n ≥ 3. Therefore, Hn−1 (Z) = ker dn−1 = Hn−1 (Yn−1 ) ⊕ F ∼
= F . As the inclusion
Yn−2 = Zn−2 → Z is an (n−2)-equivalence, we deduce that πq (Z) is trivial for 0 ≤ q ≤ n−2. In
particular, Z is simply connected as n ≥ 3. By the Hurewicz theorem, πn−1 (Z, y0 ) ∼
= Hn−1 (Z).
Choose mappings fa : S n−1 → Z, for a ∈ A, whose images under the Hurewicz mapping from a
ZM -module basis for Hn−1 (Z) ∼
= F . Then form X by attaching M × A × B n to Z = Zn−1 via
the mapping
M × A × S n−1 → Z
sending (m, a, x) to mfa (x).
Note that X is an n-dimensional CW complex with Xn−1 = Zn−1 and hence the inclusion Z = Xn−1 → X is an (n − 1)-equivalence. Therefore, πq (X) = 0 = Hq (X) for
0 ≤ q ≤ n − 2. Also πn−1 (X, y0 ) ∼
= Hn−1 (X) via the Hurewicz isomorphism. Moreover,
as the inclusion Z = Xn−1 → X is an (n − 1)-equivalence, we deduce that the inclusion
induces a surjective homomorphism πn−1 (Z, y0 ) → πn−1 (X, y0 ) and hence a surjective homomorphism Hn−1 (Z) → Hn−1 (X). As the ZM -module generators of Hn−1 (Z) have trivial images
in Hn−1 (X) by construction and the Hurewicz map, we deduce that Hn−1 (X) = 0.
Recall that Cn (X) = Hn (Xn , Xn−1 ). By standard cellular homology Hn (Xn , Xn−1 ) is a free
ZM -module on the images of the generator of the relative homology of (B n , S n−1 ) under the
characteristic mappings
ha : ({1} × {a} × B n , {1} × {a} × S n−1 ) → (Xn , Xn−1 )
and the boundary map ∂n : Hn (Xn , Xn−1 ) → Hn−1 (Xn−1 ) sends the class corresponding to
a ∈ A to the image of the generator of S n−1 under the map on homology induced by the
attaching map fa : S n−1 → Xn−1 . Hence a free basis of Hn (Xn , Xn−1 ) is sent by ∂n bijectively
to a free basis for Hn−1 (Xn−1 ) and so ∂n is an isomorphism. The long exact sequence for

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ROBERT D. GRAY AND BENJAMIN STEINBERG

reduced homology and the fact that an (n − 1)-dimensional CW complex has trivial homology
in degree n provides an exact sequence
∂

n
0 = Hn (Xn−1 ) −→ Hn (Xn ) −→ Hn (Xn , Xn−1 ) −→
Hn−1 (Xn−1 )

and so Hn (X) = Hn (Xn ) ∼
= ker ∂n = 0. As X is a simply connected n-dimensional CW
complex with Hq (X) = 0 for 0 ≤ q ≤ n, we deduce that X is contractible by the Hurewicz and
Whitehead theorems. Therefore, X is an n-dimensional equivariant classifying space for M ,
completing the proof.

We end this section by observing that monoids of left cohomological dimenison 0 are precisely
the monoids of left geometric dimension 0. The following result generalises [GP98, Lemma 1
and Theorem 1].
Proposition 6.28. Let M be a monoid. Then the following are equivalent.
(1) M has a right zero element.
(2) M has left cohomological dimension 0.
(3) M has left geometric dimension 0.
Proof. If M has a right zero z, then M z = {z} is a projective M -set and hence the one point
space is an equivariant classifying space for M . Thus M has left geometric dimension zero. If
M has left geometric dimension zero, then it has left cohomological dimension zero. If M has
left cohomological dimension zero, then Z is a projective ZM -module and so the augmentation
mapping ε : ZM → Z splits. Let P be the image of the splitting, so that ZM = P ⊕ Q. As P is
a retract of ZM and each endomorphism of ZM is induced by a right multiplication, we have
that Z ∼
= P = ZM e for some idempotent e ∈ ZM with ε(e) = 1. Then since me = e for all
m ∈ M and e has finite support X, we must have that M permutes X under left multiplication.
Let G be the quotient of M that identifies two elements if they act the same on X. Then G is
a finite group and Z must be a projective ZG-module. Therefore, G is trivial. But this means
that if x ∈ X, then mx = x for all m ∈ M and so M has a right zero.

We do not know whether left geometric dimension equals left cohomological dimension for
monoids of cohomological dimension one or two, although the former is true for groups by the
Stallings-Swan theorem.
7. Bi-equivariant classifying spaces
Let M be a monoid. We now introduce the bilateral notion of a classifying space in order
to introduce a stronger property, bi-Fn . It will turn out that bi-Fn implies both left-Fn and
right-Fn , but is strictly stronger. Moreover, bi-Fn implies bi-FPn which is of interest from the
point of view of Hochschild cohomology, which is the standard notion of cohomology for rings.
Many of the results are similar to the previous section, but the proofs are more complicated.
First recall that M is an M × M op -set via the action (mL , mR )m = mL mmR . We say that a
projective M × M op -CW complex X is a bi-equivariant classifying space for M if π0 (X) ∼
=M
as an M × M op -set and each component of X is contractible; equivalently, X has an M × M op equivariant homotopy equivalence to the discrete M × M op -set M .
We can augment the cellular chain complex of X via the canonical surjection ε : C0 (X) →
H0 (X) ∼
= Zπ0 (X) ∼
= ZM . The fact that each component of X is contractible guarantees that
this is a resolution, which will be a projective bimodule resolution of ZM and hence suitable for
computing Hochschild cohomology. We begin by establishing the uniqueness up to M × M op homotopy equivalence of bi-equivariant classifying spaces.

TOPOLOGICAL FINITENESS PROPERTIES

33

Lemma 7.1. Let X be a bi-equivariant classifying space for M and let Y be a locally path
connected M × M op -space with contractible connected components. Suppose that g : π0 (X) →
π0 (Y ) is an M ×M op -equivariant mapping. Then there exists a continuous M ×M op -equivariant
mapping f : X → Y such that the mapping f∗ : π0 (X) → π0 (Y ) induced by f is g.
Proof. Let r : X → π0 (X) and k : Y → π0 (Y ) be the projections to the set of connected
components. Then k and r are continuous M × M op -equivariant maps where π0 (X) and π0 (Y )
carry the discrete topology. Our goal will be to construct an M × M op -equivariant continuous
mapping f : X → Y such that the diagram
X

f

Y

r

π0 (X)

k

π0 (Y )

g

commutes. We construct, by induction, M ×M op -equivariant continuous mappings fn : Xn → Y
such that
Xn

fn

Y

r

π0 (X)

(7.1)

k

π0 (Y )

g

commutes and fn extends fn−1 . `
To define f0 , observe that X0 = a∈A M ea × e′a M . Choose ya ∈ Y with k(ya ) = g(r(ea , e′a )).
Then k(ea ye′a ) = ea k(ya )e′a = ea g(r(ea , e′a ))e′a = g(r(ea , e′a )) and so replacing ya by ea ye′a , we
may assume without loss of generality that ya ∈ ea Y e′a . Then by Proposition 2.3 there is an
M × M op -equivariant mapping X0 → Y given by (mea , e′a m′ ) 7→ mea ya e′a m′ for a ∈ A and
m ∈ M . By construction, the diagram (7.1) commutes.
Assume now that fn has been defined. The map k : Y → π0 (Y ) is M × M op -equivariant
and a weak equivalence (where π0 (Y ) has the discrete topology) because Y has contractible
connected components. So by Theorem 2.5 we can construct a commutative diagram
i0

Xn

Xn × I

i1

Xn

grπXn

π0 (Y )

i

k

fn

Y

Xn+1

i
fn+1

gr
i0

Xn+1 × I

i1

Xn+1

where fn+1 is M × M op -equivariant and πXn is the projection. Note that kfn+1 ≃ gr and hence,
since π0 (Y ) is a discrete space, we conclude that kfn+1 = gr. Now take f to be the colimit of
the fn . This completes the proof.

Theorem 7.2. Let X, Y be bi-equivariant classifying spaces for M . Then X and Y are M ×
M op -homotopy equivalent by a cellular M × M op -homotopy equivalence.
Proof. As π0 (X) ∼
=M ∼
= π0 (Y ) as M × M op -sets, there is an M × M op -equivariant isomorphism
g : π0 (X) → π0 (Y ). Then by Lemma 7.1, there is an M × M op -equivariant continuous mapping
f : X → Y inducing g on connected components. It follows that f is a weak equivalence as X
and Y both have contractible connected components. The result now follows from Corollary 2.7
and Theorem 2.8.


34

ROBERT D. GRAY AND BENJAMIN STEINBERG

Next we prove in an elementary fashion that bi-equivariant classifying spaces for M exist. A
more canonical construction, using simplicial sets, was described earlier.
Lemma 7.3. Let M be a monoid.
(1) If X is a projective (free) M ×M op -CW complex such that π0 (X) ∼
= M and πq (X, x) = 0
for all 1 ≤ q < n and x ∈ X, then there exists a projective M × M op -CW complex Y
containing X as a projective M × M op -CW subcomplex and such that Yn = Xn and
πq (Y, y) = 0 for all y ∈ Y and 1 ≤ q ≤ n.
(2) If X is a projective (free) M ×M op -CW complex such that π0 (X) ∼
= M and πq (X, x) = 0
for all 1 ≤ q < n and x ∈ X, then there exists a projective (free) M × M op -CW complex
Y with contractible connected components containing X as a projective M × M op -CW
subcomplex and such that Yn = Xn .
Proof. This is a minor adaptation of the proof of Lemma 6.4 that we leave to the reader.



Corollary 7.4. Let M be a monoid. Then there exists a free M × M op -CW complex X with
π0 (X) ∼
= M and each connected component of X contractible.
∼ M . We take
Proof. By Lemma 7.3 it suffices to construct a free M ×M op -graph X with π0 (X) =
X0 = M × M and we take an edge set in bijection with M × M × M . The edge (mL , mR , m) will
connect (mL , mmR ) to (mL m, mR ). Then X is a free M × M op -graph. Notice that if (m1 , m2 )
is connected by an edge to (m′1 , m′2 ), then m1 m2 = m′1 m′2 . On the other hand, (1, m2 , m1 ) is
an edge from (1, m1 m2 ) to (m1 , m2 ) and hence there is a bijection π0 (X) → M sending the
component of (m1 , m2 ) to m1 m2 and this mapping is an M × M op -equivariant bijection.

Example 7.5. It follows from the definitions and results in Section 5 that the geometric re←−→
alisation |EM | of the nerve of the two-sided Cayley graph category of M is a bi-equivariant
classifying space for M .
Corollary 7.6. If X is a bi-equivariant classifying space for M , then M \X/M ≃ |BM |.
←−→
Proof. We have M \|EM |/M ∼
= |BM |. The result now follows from Theorem 7.2 and Proposition 3.5.

Another important definition for this paper is the following. A monoid M is of type bi-Fn if
there is a bi-equivariant classifying space X for M such that Xn is M ×M op -finite, i.e., M \X/M
has finite n-skeleton. We say that M is of type bi-F∞ if M has a bi-equivariant classifying space
X that is of M × M op -finite type, i.e., M \X/M is of finite type. Clearly by making use of the
←−→
canonical two-sided classifying space |EM | we can immediately conclude that any finite monoid
is of type bi-F∞ .
Recall that a monoid M is said to be of type bi-FPn if there is a partial resolution of the
(ZM, ZM )-bimodule ZM
An → An−1 → · · · → A1 → A0 → ZM → 0
where A0 , A1 , . . . , An are finitely generated projective (ZM, ZM )-bimodules. Monoids of type
bi-FPn were studied by Kobayashi and Otto in [KO01]. We note that this differs from the
definition of bi-FPn considered in [AH03], which is called weak bi-FPn by Pride in [Pri06] where
it is shown to be equivalent to being simultaneously of type left- and right-FPn . In this paper by
bi-FPn we shall always mean bi-FPn in the above sense of Kobayashi and Otto. The property biFPn is of interest because of its connections with the study of Hochschild cohomology [Wei94,
Chapter 9]. Kobayashi investigated bi-FPn in [Kob05, Kob07, Kob10] proving, in particular,
that any monoid which admits a presentation by a finite complete rewriting system is of type

TOPOLOGICAL FINITENESS PROPERTIES

35

bi-FPn . This has applications for the computation of Hochschild cohomology. We shall recover
this theorem of Kobayashi below in Section 11 as an application of our results on equivariant
discrete Morse theory and collapsing schemes. See also [Pas08] for further related results on
bi-FPn .
The following result relates bi-Fn with bi-FPn .
Proposition 7.7. Let M be a monoid.
(1) For 0 ≤ n ≤ ∞, if M is of type bi-Fn , then it is of type bi-FPn .
(2) If M is of type bi-F∞ , then it is of type bi-Fn for all n ≥ 0.
(3) If M is of type bi-Fn for 0 ≤ n ≤ ∞, then M is of type left-Fn and type right-Fn .
(4) For 0 ≤ n ≤ ∞, a group is of type bi-Fn if and only if it is of type Fn .
Proof. The first item follows using that the cellular chain complex of a bi-equivariant classying
space X can be augmented, as discussed earlier, to give a bimodule resolution of ZM and that
if X is built up from pushouts as per (2.1) (with M × M op in place of M ), then the nth -chain
module is isomorphic to ZPn as a bimodule and hence is projective. The second item is trivial.
←−→
←−−
For the third item, one verifies that if EM is the two-sided bar construction, then EM ∼
=
←−→
←−−
EM /M where EM is the left bar construction. Suppose now that X is a bi-equivariant
←−→
classifying space for M such that Xn is M × M op -finite. Then X ≃M ×M op EM by Theo←−−
←−→
rem 7.2. Therefore, X/M ≃M |EM |/M = |EM | and X/M is a projective M -CW complex with
(X/M )n = Xn /M being M -finite by Corollary 3.8. Thus if M is of type bi-Fn for 0 ≤ n ≤ ∞,
then M is of type left-Fn and dually right-Fn .
If G is a group of type bi-Fn , then it is of type Fn by the previous item and Proposition 6.9.
Conversely, suppose that X is a free G-CW complex with G-finite n-skeleton. Then using the
right G-set structure on G × G from Proposition 3.7 we have that Y = (G × G) ⊗G X is a
projective G × Gop -CW complex by Proposition 3.2 such that Yn is G × Gop -finite. Moreover,
π0 (Y ) ∼
= (G×G)⊗G π0 (X) by Proposition 3.4. But π0 (X) is the trivial G-set and (G×G)⊗G 1 ∼
=
G as a G × Gop -set via (g, h) ⊗ 1 7→ gh. Finally, since G is a free
right
G-set
of
on
G-generators
`
by Proposition 3.7, it follows that as a topological space Y = G X and hence each component
of Y is contractible. This completes the proof.

The proof of Proposition 7.7 establishes the following proposition.

Proposition 7.8. If X is a bi-equivariant classifying space for M , then X/M is an equivariant
classifying space for M .
Sometimes it will be convenient to use the following reformulation of the property bi-Fn .
Proposition 7.9. Let M be a monoid. The following are equivalent for 0 ≤ n < ∞.
(1) M is of type bi-Fn
(2) There is an M × M op -finite projective M × M op -CW complex X of dimension at most
n with π0 (X) ∼
= M and πq (X, x) = 0 for 1 ≤ q < n and x ∈ X.
Proof. This is entirely analogous to the proof of Proposition 6.10.


←−−−−→
If M is a monoid and A ⊆ M , then the two-sided Cayley digraph Γ(M, A) is the digraph
with vertex set M × M and with edges in bijection with elements of M × M × A. The directed
edge (mL , mR , a) goes from (mL , amR ) to (mL a, mR ) and we draw it as
a

(mL , amR ) −→ (mL a, mR ).
←−−−−→
Note that Γ(M, A) is a free M × M op -graph and is M × M op -finite if and only if A is finite.
Also note that if (m1 , m2 ) is connected to (m′1 , m′2 ) by an edge, then m1 m2 = m′1 m′2 . Hence

36

ROBERT D. GRAY AND BENJAMIN STEINBERG

multiplication of the coordinates of a vertex induces a surjective M × M op -equivariant mapping
←−−−−→
π0 (Γ(M, A)) → M . If A is a generating set for M , then the mapping is an isomorphism because
if m1 , m2 ∈ M and u ∈ A∗ is a word representing m1 , then there is a directed path labelled by
u from (1, m) to (m1 , m2 ). Namely, if u = a1 · · · ak with ai ∈ A, then the path labelled by u
from (1, m) to (m1 , m2 ) is
a

a

a

a

(1, m) −−1→ (a1 , a2 · · · ak m2 ) −−2→ (a1 a2 , a3 · · · ak m2 ) −−3→ · · · −−k→ (m1 , m2 ).

(7.2)

A monoid M is said to be dominated by a subset A if whenever f, g : M → N are monoid
homomorphisms with f |A = g|A , one has f = g. In other words, the inclusion hAi ֒→ M is
an epimorphism (in the category theory sense). Of course, a generating set of M dominates
M . Note that if A is a subset of an inverse monoid M (e.g., a group), then A dominates M if
and only if A generates M as an inverse monoid. Hence M is finitely generated if and only if
M is finitely dominated. Kobayashi gives an example of an infinitely generated monoid that is
finitely dominated. See [Kob07] for details.
Theorem 7.10. The following are equivalent for a monoid M .
(1) M is of type bi-F1 .
(2) M is of type bi-FP1 .
←−−−−→
(3) There is a finite subset A ⊆ M such that the natural mapping π0 (Γ(M, A)) → M is an
isomorphism.
(4) There is a finite subset A ⊆ M that dominates M .
In particular, any finitely generated monoid is of type bi-F1 .
Proof. The equivalence of (2) and (4) was established by Kobayashi [Kob07] using Isbel’s zig-zag
lemma (actually, the equivalence of (3) and (4) is also direct from Isbel’s zig-zag lemma).
←−−−−→
Assume that (3) holds. Then Γ(M, A) is M × M op -finite and so M is of type bi-F1 by
Proposition 7.9. Proposition 7.7 shows that (1) implies (2).
Assume that M is of type bi-FP1 . Then we have a partial free resolution
µ

ZM ⊗ ZM −−→ ZM −→ 0
of finite type where µ is induced by the multiplication in ZM . Since M is of type bi-FP1 , ker µ
op -module by the
is finitely generated. It is well known that ker µP
is generated as a ZM ⊗ ZMP
elements m ⊗ 1 − 1 ⊗ m with m ∈ M . Indeed, if
ci mi ⊗ ni is in ker µ, then
ci mi ni = 0 and
so
X
X
X
ci mi ⊗ ni =
ci mi ⊗ ni −
ci (1 ⊗ mi ni )
X
=
ci (mi ⊗ ni − 1 ⊗ mi ni )
X
=
ci (mi ⊗ 1 − 1 ⊗ mi )ni .

Hence there is a finite subset A ⊆ M such that ker µ is generated by the elements a⊗1−1⊗a with
←−−−−→
a ∈ A. We claim that the natural surjective mapping π0 (Γ(M, A)) → M is an isomorphism.
Identifying Z[M ×M op ] with ZM ⊗ZM op as rings and Z[M ×M ] with ZM ⊗ZM as bimodules,
we have a bimodule homomorphism
←−−−−→
λ : Z[M × M ] → Zπ0 (Γ(M, A))
←−−−−→
sending (mL , mR ) to its connected component in Γ(M, A) and µ factors as λ followed by the
natural mapping
←−−−−→
Zπ0 (Γ(M, A)) → ZM.

TOPOLOGICAL FINITENESS PROPERTIES

37

Clearly, ker λ ⊆ ker µ and so to prove the result it suffices to show that ker µ ⊆ ker λ. Now ker µ
is generated by the elements (1, a) − (a, 1) with a ∈ A under our identifications. But (1, 1, a) is
an edge from (1, a) to (a, 1). Thus (1, a) − (a, 1) ∈ ker λ for all a ∈ A. This establishes that (2)
implies (3), thereby completing the proof.

If G is a group, then it follows from Theorem 7.10 that G ∪ {0} is of type bi-F1 if and only
if G is finitely generated. Indeed, G ∪ {0} is an inverse monoid and hence finitely dominated
if and only if finitely generated. But G ∪ {0} is finitely generated if and only if G is finitely
generated. On the other hand, G ∪ {0} is both of type left- and right-F∞ for any group G by
Corollary 6.23. Thus bi-Fn is a much stronger notion.
Remark 7.11. It can be shown that if M is a monoid and M 0 is the result of adjoining a 0 to M ,
then if M 0 is of type bi-Fn , then M is of type bi-Fn . The idea is that if X is a bi-equivariant
classifying space for M 0 , then the union Y of components of X corresponding to elements of M
is a bi-equivariant classifying space for M and Yn will be M × M op -finite if Xn is M 0 × (M 0 )op finite. More generally, if T is a submonoid of M such that M \ T is an ideal, then M being of
type bi-Fn implies T is also of type bi-Fn .
Next we show that finitely presented monoids are of type bi-F2 . The proof is similar to the
proof of Theorem 6.14, which is in fact a consequence.
Theorem 7.12. Let M be a finitely presented monoid. Then M is of type bi-F2 .
Proof. Suppose that M is generated by a finite set A with defining relations u1 = v1 , . . . , un =
vn . We construct an M × M op -finite 2-dimensional free M × M op -CW complex X with 1←−−−−→
skeleton the two-sided Cayley graph Γ(M, A) by attaching an M × M op -cell M × M × B 2 for
each relation. Suppose that ui and vi map to mi in M . Let pi , qi be the paths from (1, mi ) to
(mi , 1) labelled by ui and vi , respectively, cf. (7.2). Then we glue in a disk di with boundary
path pi qi−1 and glue in M × M × B 2 using Proposition 2.3 (so {(mL , mR )} × B 2 is sent to
mL di mR ). Then X is a free M × M op -CW complex of dimension at most 2 that is M × M op finite and π0 (X) ∼
= M . By Proposition 6.10, it suffices to prove that each connected component
of X is simply connected.
The connected component X(m) of X corresponding to m ∈ M is a digraph rooted at (1, m)
a
by (7.2). Let Tm be a directed spanning tree for X(m) rooted at (1, m). Let e = (n1 , an2 ) −→
(n1 a, n2 ) be a directed edge of X(m) not belonging to Tm . Then the corresponding generator of
π1 (X(m), (1, m)) is of the form peq −1 where p and q are directed paths from (1, m) to (n1 , an2 )
and (n1 a, n2 ), respectively. Let u be the label of p and v be the label of q. Then ua = v in
M . Thus it suffices to prove that if x, y ∈ A∗ are words which are equal in M to m′ labelling
respective paths from (1, m) to (m′ , m′′ ) with m′ m′′ = m, then the corresponding loop ℓ labelled
xy −1 at (1, m) is null homotopic.
By induction on the length of a derivation from x to y, we may assume that x = wui w′
and y = wvi w′ for some i = 1, . . . , n. Then the path labelled by w starting at (1, m) ends at
(w, mi w′ m′′ ) where we recall that mi is the image of ui , vi in M . Then wdi w′ m′′ is a 2-cell
bounded by parallel paths from (w, mi w′ m′′ ) to (wmi , w′ m′′ ) labeled by ui and vi , respectively.
It follows that the paths labelled by x and y from (1, m) to (m′ , m′′ ) are homotopic relative
to endpoints and hence ℓ is null homotopic. This completes the proof that X(m) is simply
connected.

Remark 7.13. We currently do not know the precise relationship between bi-F2 and finitely
presentability for monoids. Specifically we have the question: Is there a finitely generated bi-F2
monoid that is not finitely presented? Even for inverse monoids this question remains open.

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ROBERT D. GRAY AND BENJAMIN STEINBERG

We next observe that finitely generated free monoids are bi-F∞ .
Proposition 7.14. Let A be a finite set. Then the free monoid A∗ is of type bi-F∞ .
←−−−−→
←−−−−→
Proof. Each connected component of Γ(M, A) is a tree and hence contractible. Thus Γ(M, A)
is an A∗ × (A∗ )op -finite bi-equivariant classifying space for A∗ .

Theorem 6.15 has an analogue for bi-Fn and bi-FPn with essentially the same proof, which
we omit.
Theorem 7.15. Let M be a monoid of type bi-F2 . Then M is of type bi-Fn if and only if M
is of type bi-FPn for 0 ≤ n ≤ ∞.
Observe that Theorem 7.15 implies that M is of type bi-F∞ if and only if M is of type bi-Fn
for all n ≥ 0. The analogue of Proposition 6.16 in our setting again admits a very similar proof
that we omit.
Proposition 7.16. If M is of type bi-Fn with n ≥ 1, then M has a free M × M op -CW complex
X that is a bi-equivariant classifying space for M where Xn is M × M op -finite.
Proposition 6.26 also has a two-sided analogue.
Proposition 7.17. If M, N are of type bi-Fn , then M × N is of type bi-Fn .
Let us turn to some inheritance properties for bi-Fn . If I is an ideal of M containing an
identity e, then e is a central idempotent and M eM = M e = eM = eM e. Indeed, em =
(em)e = e(me) = me as em, me ∈ I. If f, f ′ ∈ E(M ), then f e, f ′ e ∈ E(eM e) and e(M f ×
f ′ M )e = eM ef e × f ′ eM e as an eM e-eM e-biset and hence is finitely generated projective.
Thus if P is a (finitely generated) projective M × M op -set, then eP e is a (finitely generated)
projective eM e × eM eop -set.
Proposition 7.18. Let M be a monoid and 0 ≤ n ≤ ∞ and e ∈ E(M ) be a central idempotent.
If M is of type bi-Fn , then so is eM e.
Proof. Let X be a bi-equivariant classifying space of M such that Xn is M ×M op -finite. Suppose
that X is obtained via pushouts as per (2.1) (but with M ×M op in place of M ). Then each ePk e
is a projective eM e × eM eop -set and is finitely generated whenever Pk was finitely generated
by the observation preceding the proposition. Thus eXe is a projective eM e × eM eop -CW
complex and (eXe)n = eXn e is eM e × eM eop -finite. Also, since eXe ∼
= (eM × M e) ⊗M ×M op X,
∼
∼
we deduce that π0 (eXe) = eπ0 (X)e = eM e by Proposition 3.4. If X(m) is the component of X
corresponding to m ∈ eM e, then eX(m)e is the component of eXe corresponding to m in eXe
and is a retract of X(m). But X(m) is contractible and hence eX(m)e is contractible. This
shows that eXe is a bi-equivariant classifying space for eM e, completing the proof.

Two monoids M and N are Morita equivalent if the categories of left M -sets and left N -sets
are equivalent. It is known that this is the case if and only if there is an idempotent e ∈ E(M )
such that xy = 1 for some x, y ∈ M with ey = y and eM e ∼
= N [Kna72]. It follows easily
that if M and N are Morita equivalent, then so are M op and N op . Note that if e is as above,
then the functor A 7→ eA ∼
= eM ⊗M A from M -sets to N -sets (identifying N with eM e) is an
equivalence of categories with inverse B 7→ M e ⊗N B. This uses that M e ⊗eM e eM ∼
= M as
op
M × M -sets via the multiplication map (the inverse bijection takes m ∈ M to mxe ⊗ y) and
eM ⊗M M e ∼
= eM e as eM e × (eM e)op -sets (via the multiplication with inverse eme 7→ em ⊗ e).
It follows that if P is a (finitely generated) projective M -set, then eP is a (finitely generated)
projective N -set (as being projective is categorical and a projective is finitely generated if and
only if it is a coproduct of finitely many indecomposable projectives, which is also categorical).
In particular, eM is a finitely generated projective N -set.

TOPOLOGICAL FINITENESS PROPERTIES

39

Proposition 7.19. Let M and N be Morita equivalent monoids and 0 ≤ n ≤ ∞.
(1) M is of type left-Fn if and only if N is of type left-Fn .
(2) M is of type right-Fn if and only if N is of type right-Fn .
(3) M is of type bi-Fn if and only if N is of type bi-Fn .
Proof. By symmetry it suffice to prove the implications from left to right. We may assume
without loss of generality that N = eM e where 1 = xy with ey = y. Notice that 1 = xy = xey
and so replacing x by xe, we may assume that xe = x. To prove (1), suppose that X is an
equivariant classifying space for M such that Xn is M -finite. Then eM ⊗M X ∼
= eX is a
projective N -CW complex by Proposition 3.2 such that (eX)n = eXn is N -finite. But eX is a
retract of X and hence contractible. We deduce that N is of type left-Fn .
The proof of (2) is dual. To prove (3), observe that (e, e)(M × M op )(e, e) = eM e × (eM e)op
and that we have (x, y)(y, x) = (1, 1) and (e, e)(y, x) = (y, x) in M ×M op because xy = e, ey = y
and xe = x. Thus M ×M op is Morita equivalent to N ×N op and eM ×M e is a finitely generated
projective N × N op -set. Suppose that X is a bi-equivariant classifying space for M such that
Xn is M ×M op -finite. Then (eM ×M e)⊗M ×M op X ∼
= eXe is a projective N ×N op -CW complex
such that (eXe)n = eXn e is N × N op -finite by Corollary 3.2. Also, π0 (eXe) ∼
=N
= eπ(X)e ∼
by Proposition 3.4. Moreover, if m ∈ eM e and X(m) is the component of X corresponding to
m, then the component of eXe corresponding to m is eX(m)e, which is a retract of X(m) and
hence contractible. Thus eXe is a bi-equivariant classifying space of N . The result follows. 
There are examples of Morita equivalent monoids that are not isomorphic; see [Kna72].
We define the geometric dimension of M to be the minimum dimension of a bi-equivariant
classifying space for M . The Hochschild cohomological dimension of M , which we write dim M ,
is the length of a shortest projective resolution of ZM as a Z[M × M op ]-module. Of course,
the Hochschild cohomological dimension bounds both the left and right cohomological dimension and the geometric dimension bounds the Hochschild cohomological dimension. Also the
geometric dimension bounds both the left and right geometric dimensions because if X is a biequivariant classifying space for M of dimension n, then X/M is a classifying space of dimension
n.
The following theorem has an essentially identical proof to Theorem 6.27.
Theorem 7.20. Let M be a monoid. Then M has a bi-equivariant classifying space of dimension max{dim M, 3}.
Free monoids have a forest for a bi-equivariant classifying space and hence have geometric
dimension 1. It is well known (see e.g. [Mit72]) that they have Hochschild cohomological
dimension 1.
It is known that a monoid has Hochschild cohomological dimension 0 if and only if it is a
finite regular aperiodic monoid with sandwich matrices invertible over Z (see [Che84]). For
instance, any finite aperiodic inverse monoid has Hochschild cohomogical dimension 0. A nontrivial monoid of Hochschild cohomological dimension 0 does not have geometric dimension 0
because M would have to be a projective M -biset. So M ∼
= M e × f M , with e, f ∈ E(M ),
via an equivariant map ϕ sending (e, f ) to 1 (as M being finite aperiodic implies that 1 is the
unique generator of M as a two-sided ideal). But then f = ϕ(e, f )f = ϕ(e, f ) = 1 and similarly
e = 1 and so M is trivial. Thus non-trivial monoids of Hochschild cohomological dimension 0
do not have geometric dimension 0.

40

ROBERT D. GRAY AND BENJAMIN STEINBERG

8. Brown’s theory of collapsing schemes
The theory of collapsing schemes was introduced by Brown in [Bro92]. Since then it has
become an important and often-used tool for proving that certain groups are of type F∞ . The
first place the idea appears is in a paper of Brown and Geoghegan [BG84] where they had a
cell complex with one vertex and infinitely many cells in each positive dimension, and they
showed how it could be collapsed to a quotient complex with only two cells in each positive
dimension. Brown went on to develop this idea further in [Bro92] formalising it in his theory
of collapsing schemes, and applying it to give a topological proof that groups which admit
presentations by finite complete rewriting systems are of type F∞ (see Section 11 below for
the definition of complete rewriting system). Brown’s theory of collapsing schemes was later
rediscovered under the name of discrete Morse theory [For95, For02], an important area in
algebraic combinatorialists. Chari [Cha00] formulated discrete Morse theory combinatorially
via Morse matchings, which turn out to be the same thing as collapsing schemes.
The basic idea of collapsing schemes for groups is a follows. Suppose we are given a finitely
presented group G and we would like to prove it is of type F∞ . Then we can first begin with the
big K(G, 1) complex |BG| with infinitely many n-cells for each n. Then in certain situations
it is possible to show how one can collapse away all but finitely many cells in each dimension
resulting in a K(G, 1) much smaller than the one we started with. The collapse is carried out
using a so-called collapsing scheme associated with the simplicial set BG. It turns out that any
group which is presentable by a finite complete rewriting system admits a collapsing scheme
that, using this process, can be used to prove the group is of type F∞ ; see [Bro92, page 147].
As mentioned in the introduction above, Brown in fact develops this theory for monoids in
general, and applies the theory of collapsing schemes to show that if M admits a presentation
by a finite complete rewriting system then its classifying space |BM | has the homotopy type of
a CW complex with only finitely many cells in each dimension. However, as discussed in detail
in the introduction to this article, this information about the space |BM | is not enough on its
own to imply that the monoid M is of type left-FP∞ .
Motivated by this, in this section we shall develop the theory of M -equivariant collapsing
schemes. We shall prove that if an M -simplicial set admits an M -equivariant collapsing scheme
of finite type then the monoid is of type left-F∞ . We then prove that if M admits a finite
−−→
complete rewriting system then EM admits an M -equivariant collapsing scheme of finite type,
thus giving a topological proof that such monoids are of type left-F∞ . To do this, we shall identify conditions under which a collapsing scheme for BM can be lifted to give an M -equivariant
−−→
collapsing scheme for EM . These conditions will hold in a number of different situations, including when M admits a presentation by a finite complete rewriting system and when M is a,
so-called, factorable monoid [HO14]. We also develop the two-sided theory. As a consequence
we also obtain a topological proof of the fact that such a monoid is of type bi-F∞ , recovering
a theorem of Kobayashi [Kob05].
S
8.1. Collapsing schemes. Let K = i≥0 Ki be a simplicial set and let X = |K| be its
geometric realisation. We identify the cells of X with the non-degenerate simplices of K. A
collapsing scheme for K consists of the following data:
• A partition of the cells of X into three classes, E, C, R, called the essential, collapsible
and redundant cells, respectively, where the collapsible cells all have dimension at least
one.
• Mappings c and i which associate with each redundant n-cell τ a collapsible (n + 1)-cell
c(τ ), and a number i(τ ), such that τ = di(τ ) (c(τ )).

TOPOLOGICAL FINITENESS PROPERTIES

41

Let σ = c(τ ). If τ ′ is a redundant n-cell such that τ ′ = dj σ for some j 6= i(τ ) then we call τ ′
an immediate predecessor of τ and write τ ′ ≺ τ . Furthermore, the conditions for a collapsing
scheme are satisfied, which means:
(C1) for all n, the mapping c defines a bijection between Rn (the redundant n-cells) and
Cn+1 (the collapsible (n + 1)-cells).
(C2) there is no infinite descending chain τ ≻ τ ′ ≻ τ ′′ ≻ · · · of redundant n-cells.
Condition (C2) clearly implies that there is a unique integer i such that τ = di (c(τ )) (otherwise
we would have τ ≻ τ , leading to an infinite descending chain). It also follows from (C2) that,
by Königs lemma, there cannot be arbitrarily long descending chains τ0 ≻ · · · ≻ τk . This is a
key fact in the proof of [Bro92, Proposition 1] since it gives rise to the notion of ‘height’:
Definition 8.1 (Height). The height of a redundant cell τ , written height(τ ), is the maximum
length of a descending chain τ = τ0 ≻ τ1 ≻ · · · ≻ τk .
We say that a collapsing scheme is of finite type if it has finitely many essential cells of each
dimension.
In the construction of the ‘small’ CW complex in the proof of [Bro92, Proposition 1] the
redundant n-cells are adjoined in order of their heights, guaranteeing in the proof that the
adjunction of τ and c(τ ) is an elementary expansion. This is the the key idea in the proof
which is that each pair (τ, c(τ )) of redundant and corresponding collapsible cells may be adjoined
without changing the homotopy type, and so in the end it is only the essential cells that matter.
More precisely, Brown proves that if K be a simplical set with a collapsing scheme, then its
geometric realisation X = |K| admits a canonical quotient CW complex Y , whose cells are
in 1–1 correspondence with the essential cells of X. This notion of height in Brown’s theory
relates to the values taken by the discrete Morse function in Forman’s theory (see [For02, page
10]). A discrete Morse function gives one a way to build the simplicial complex by attaching
the simplices in the order prescribed by the function, i.e., adding first the simplices which are
assigned the smallest values. Brown’s essential cells are called ‘critical’ in Forman’s theory.
9. M -equivariant collapsing schemes
In this section we develop the theory of M -equivariant collapsing schemes, or equivalently,
of M -equivariant discrete Morse theory. Results on G-equivariant discrete Morse theory, for G
a group, may
S be found in [Fre09].
Let K = i≥0 Ki be a simplicial set with degeneracy and face operators di , si , and equipped
with a collapsing scheme (E, R, C, c, i). Here E, R and C partition the cells (which are in
bijective correspondence with the non-degenerate simplices) of K.
Let M be a monoid acting on the simplicial set K with the following conditions satisfied:
(A1) The action of M maps n-simplicies to n-simplicies, and commutes with di and si , that
is, M is acting by simplicial morphisms.
(A2) For every n-simplex σ and m ∈ M , σ is a cell (i.e. is a non-degenerate simplex) if and
only if mσ is a cell, in which case σ ∈ E (respectively R, C) if and only if mσ ∈ E
(respectively R, C).
(A3) If (σ, τ ) ∈ Rn × Cn+1 is a matched redundant-collapsible pair (i.e. τ = c(σ)) then so is
the pair m(σ, τ ) = (mσ, mτ ) ∈ Rn × Cn+1 , i.e., c(mσ) = mc(σ) for σ ∈ Rn .
(A4) There is a subset B ⊆ E ∪ R ∪ C such that for all n the set of n-cells is a free left M -set
with basis Bn (where Bn is the set of n-cells in B). Let E B = E ∩ B, RB = R ∩ B and
C B = C ∩ B. Then En is a free left M -set with basis EnB , and similarly for Rn and Cn .
(A5) For every matched pair (σ, τ ) ∈ R × C, σ ∈ RB if and only if τ ∈ C B . In particular,
for every matched pair (σ, τ ) there is a unique pair (σ ′ , τ ′ ) ∈ RB × C B and m ∈ M

42

ROBERT D. GRAY AND BENJAMIN STEINBERG

such that (σ, τ ) = m(σ ′ , τ ′ ); namely, if σ = mσ ′ with σ ′ ∈ RB and τ ′ = c(σ ′ ), then
mτ ′ = mc(σ ′ ) = c(mσ ′ ) = c(σ) = τ .
(A6) For every redundant cell τ and every m ∈ M
height(τ ) = height(mτ ),
with height defined as in Definition 8.1 above.
These conditions imply that K is a rigid free left M -simplicial set and hence by Lemma 4.4 the
action of M on K induces an action of M on the geometric realisation |K| by continuous maps
making |K| into a free left M -CW complex. When the above axioms hold, we call (E, R, C, c, i)
an M -equivariant collapsing scheme for the rigid free left M -simplicial set K. Dually, given a
rigid free right M -simplicial set K with a collapsing scheme satisfying the above axioms for K
as an M op -simplicial set we call (E, R, C, c, i) an M -equivariant collapsing scheme for K. If K
is a bi-M -simplicial set we say (E, R, C, c, i) an M -equivariant collapsing scheme if the axioms
are satisfied for K as a left M × M op -simplicial set.
Our aim is to prove a result about the M -homotopy type of |K| when K has an M -equivariant
collapsing scheme. Before doing this we first make some observations about mapping cylinders
and the notion of elementary collapse.
9.1. Mapping cylinders and elementary collapse. If X is a subspace of a space Y then
D : Y → X is a strong deformation retraction if there is a map F : Y × I → Y such that, with
Ft : Y → Y defined by Ft (y) = F (y, t), we have (i) F0 = 1Y , (ii) Ft (x) = x for all (x, t) ∈ X × I,
and (iii) F1 (y) = D(y) for all y ∈ Y . If D : X → Y is a strong deformation retraction then D
is a homotopy equivalence, a homotopy inverse of which is the inclusion i : X ֒→ Y .
Definition 9.1 (Mapping cylinder). Let f : X → Y be a cellular map
`between CW complexes.
The mapping cylinder Mf is defined to be the adjunction complex Y f0 (X × I) where f0 : X ×
{0} is the map (x, 0) 7→ f (x). Let i1 : X → X × I, x 7→ (x, 1) and let i0 : X → X × I, x 7→ (x, 0).
Let p be the projection p : X × I → X, (x, i) 7→ x. Also set i = k ◦ i1 , with k as below. Thus
we have
X
p

f

Y
j

i1 i0

X ×I

k

Mf

The map from X × I to Y taking (x, t) to f (x), and the identity map on Y , together induce a
retraction r : Mf → Y .
The next proposition is [Geo08, Proposition 4.1.2].
Proposition 9.2. The map r is a homotopy inverse for j, so r is a homotopy equivalence.
Indeed there is a strong deformation retraction D : Mf × I → Mf of Mf onto Y such that
D1 = r.
The following result is the M -equivariant analogue of [Geo08, Proposition 4.1.2]. Recall that
if X is a projective (resp. free) M -CW complex then Y = M × I is a projective (resp. free)
M -CW complex, where I is given the trivial action.

TOPOLOGICAL FINITENESS PROPERTIES

43

Lemma 9.3. Let f : X → Y be a continuous M -equivariant cellular map between free (projective) M -CW complexes X and Y . Let Mf be the pushout of
X

f

Y
j

i0

X ×I

k

Mf

where i0 : X → X × I, x 7→ (x, 0). Then:
(i) Mf is a free (projective) M -CW complex; and
(ii) there is an M -equivariant strong deformation retraction r : Mf → Y . In particular Mf
and Y are M -homotopy equivalent.
Proof. It follows from Lemma 2.2 that Mf has the structure of a free (projective) M -CW
complex, proving part (i). For part (ii) first note that the map from X × I to Y taking (x, t)
to f (x), and the identity map on Y , together induce a retraction r : Mf → Y . It follows from
Proposition 9.2 that r is a homotopy equivalence with homotopy inverse j. By Corollary 2.7 to
show that r is an M -homotopy equivalence it suffices to verify that r is an M -equivariant map
between the sets Mf and Y . But M -equivariance of r follows from the definitions of r and Mf ,
the definition of the action of M on Mf which in turn is determined by the actions of M on
X × I and on Y , together with the assumption that f : X → Y is M -equivariant.

The fundamental idea of collapsing schemes, and discrete Morse theory, is that of a collapse.
The following definition may be found in [Coh73, page 14] and [FS05, Section 2]. We use the
same notation as in [Coh73]. In particular ≈ denotes homeomorphism of spaces.
Definition 9.4 (Elementary collapse). If (K, L) is a CW pair then K collapses to L by an
elementary collapse, denoted K ցe L, if and only if:
(1) K = L∪en−1 ∪en where en and en−1 are open cells of dimension n and n−1 respectively,
which are not in L, and
(2) there exists a ball pair (B n , B n−1 ) ≈ (I n , I n−1 ) and a map ϕ : B n → K such that
(a) ϕ is a characteristic map for en
(b) ϕ|B n−1 is a characteristic map for en−1
(c) ϕ(P n−1 ) ⊆ Ln−1 where P n−1 = cl(∂B n − B n−1 ).
Note that in this statement P n−1 is an (n − 1)-ball (i.e. is homeomorphic to I n−1 ). We
say that K collapses to L, writing K ց L, if L may be obtained from K by a sequence of
elementary collapses. We also say that K is an elementary expansion of L. An elementary
collapse gives a way of modifying a CW complex K, by removing the pair {en−1 , en }, without
changing the homotopy type of the space. We can write down a homotopy which describes such
an elementary collapse K ցe L as follows. Let (K, L) be a CW pair such that K ցe L. Set
ϕ0 = ϕ|pn−1 in the above definition. Then
ϕ0 : (P n−1 , ∂P n−1 ) → (Ln−1 , Ln−2 ),
(using the identification P n−1 ≈ I n−1 ) and
(K, L) ≈ (L

a

B n , L).

ϕ0

The following is [Coh73, (4.1)].
Lemma 9.5. If K ցe L then there is a cellular strong deformation retraction D : K → L.

44

ROBERT D. GRAY AND BENJAMIN STEINBERG

Indeed, let K = L ∪ en−1 ∪ en . There is a map ϕ0 : I n−1 ≈ P n−1 → Ln−1 such that
a
B n , L).
(K, L) ≈ (L
ϕ0

`

But L ϕ0 B n is the mapping cylinder Mϕ0 of ϕ0 : I n−1 → Ln−1 . Thus by Proposition 9.2 there
is a strong deformation retraction
a
In → L
D: K ≈ L
ϕ0

such that D(en ) = ϕ0 (I n−1 ) ⊂ Ln−1 . The map D is given by the map r in Definition 9.1.
We may now state and prove the main result of this section.

Theorem 9.6. Let K be a rigid free left M -simplicial set with M -equivariant collapsing scheme
(E, R, C, c, i) (that is, the conditions (A1)–(A6) are satisfied). Then, with the above notation,
there is a free left M -CW complex Y with Y ≃M |K| and such that the cells of Y are in bijective
correspondence with E, and under this bijective correspondence Yn is a free left M -set with basis
EnB for all n.
Proof. Let X be the geometric realisation |K| of the simplicial set K. By axiom (A1) we have
that K is a left M -simplical set, and it follows that X = |K| has the structure of a left M -CW
complex where the M -action is given by Lemma 4.4. In fact, by assumptions (A2)-(A6), X is
a free M -CW complex where, for each n, the set EnB ∪ RnB ∪ CnB is a basis for the n-cells.
Write X as an increasing sequence of subcomplexes
X0 ⊆ X0+ ⊆ X1 ⊆ X1+ ⊆ . . .
where, X0 consists of the essential vertices, Xn+ is obtained from Xn by adjoining the redundant
n-cells and collapsible (n + 1)-cells, and Xn+1 is obtained from Xn+ by adjoining the essential
(n + 1)-cells. We write Xn+ as a countable union
Xn = Xn0 ⊆ Xn1 ⊆ Xn2 ⊆ . . .

`
with Xn+ = i≥0 Xni where Xnj+1 is constructed from Xnj by adjoining (τ, c(τ )) for every redundant n-cell τ of height j. From assumptions (A1)-(A6), for every n and j, each of Xn+ , Xn and
Xnj is a free M -CW subcomplex of X.
As argued in the proof of [Bro92, Proposition 1], for every redundant n-cell τ of height j the
adjunction of (τ, c(τ )) is an elementary expansion. In this way Xnj+1 can be obtained from Xnj
by a countable sequence of simultaneous elementary expansions. The same idea, together with
Lemma 9.3, can be used to obtain an M -homotopy equivalence between Xnj and Xnj+1 . The
details are as follows.
Recall that Xnj+1 is obtained from Xnj by adjoining (τ, c(τ )) for every redundant n-cell τ of
height j. It follows from the axioms (A1)-(A6) that this set of pairs (τ, c(τ )) is a free M -set with
B
B
with height(τ ) = j,
: height(τ ) = j}. Let (τ, c(τ )) ∈ RnB × Cn+1
basis {(τ, c(τ )) ∈ RnB × Cn+1
and let m ∈ M . From the assumptions (A1)-(A6) it follows that
m · (τ, c(τ )) = (mτ, c(mτ )),

and

height(mτ ) = height(τ ) = j.

The pair
(Xnj+1 , Xnj+1 − {mτ, c(mτ )})
satisfies the conditions of an elementary collapse. Indeed (as argued in the proof of [Bro92,
Proposition 1]) every face of c(mτ ) other than mτ is either (i) a redundant cell of height
< j, (ii) is essential (so has height 0), or (iii) is collapsible or degenerate. It follows that the
adjunction of mτ and c(mτ ) is an elementary expansion. This is true for every pair (mτ, c(mτ ))

TOPOLOGICAL FINITENESS PROPERTIES

45

B
with height(τ ) = j and m ∈ M . Now Xnj+1 is obtained from Xnj
where (τ, c(τ )) ∈ RnB × Cn+1
B
: height(τ ) = j} and
by adjoining all such pairs (mτ, c(mτ )). Let A = {(τ, c(τ )) ∈ RnB × Cn+1
let M × A denote the free left M -set with basis {(1, a) : a ∈ A} and action m(n, a) = (mn, a).
Then (M × A) × I n+1 is a disjoint union of the free M -cells (M × {a}) × I n+1 with a ∈ A. The
characteristic maps for the collapsible n + 1 cells of height j combine to give an M -equivariant
map ϕ : (M × A) × I n+1 → Xnj+1 such that
(E1) ϕ restricted to the (m, a) × I n+1 , (m ∈ M, a ∈ A), gives characteristic maps for each of
the collapsible n + 1 cells of height j;
(E2) ϕ restricted to the (m, a) × I n , (m ∈ M, a ∈ A), gives characteristic maps for each
τ ∈ Rn such that c(τ ) is a collapsible n + 1 cell of height j;
(E3) ϕ((M × A) × P n ) ⊆ (Xnj+1 )≤n (where (Xnj+1 )≤n is the subcomplex of Xnj+1 of cells of
dimension ≤ n) where P n = cl(∂I n+1 − I n ).
Set ϕ0 = ϕ|(M ×A)×P n . Then

ϕ0 : (M × A) × P n → (Xnj+1 )≤n
is a continuous M -equivariant cellular map between free M -CW complexes (M × A) × P n and
(Xnj+1 )≤n . It follows that Xnj+1 is M -equivariantly isomorphic to
a
(9.1)
(Xnj+1 − {(τ, c(τ )) ∈ Rn × Cn+1 : height(τ ) = j}) ((M × A) × I n+1 ).
ϕ0

Pn

cl(∂I n+1

− I n)

In

But since
=
≈
we conclude that (9.1) is just the mapping cylinder of ϕ0 .
Thus we can apply Lemma 9.3 to obtain a strong deformation retraction rj : Xnj+1 → Xnj which
is also an M -homotopy equivalence. It follows that there is a retraction rn : Xn+ → Xn which
is an M -equivariant homotopy equivalence.
We build the space Y such that |K| ≃M Y inductively. First set Y0 = X0 . Now suppose
+
that we have an M -homotopy equivalence π n−1 : Xn−1
→ Yn−1 is given. Define Yn to be the
B
B
the M -CW complex Yn−1 ∪ (M × En ) where (M × En ) is a collection of free M -cells indexed
by EnB . These free M -cells are attached to Yn by composing with π n−1 the attaching maps
+
for the essential n-cells of X. This makes sense because Xn−1
contains the (n − 1)-skeleton
n−1
n−1 : X → Y in the obvious way. This
of X. Extend π
to an M -homotopy equivalence π[
n
n
+
is possible since Xn is obtained from Xn−1 by adjoining the essential n-cells. Composing rn
n−1 then gives an M -homotopy equivalence X + → Y . Passing to the union gives the
with π[
n
n
M -homotopy equivalence X ≃M Y stated in the theorem.

There is an obvious dual result for right simplicial sets which follows from the above result
by replacing M by M op . We also have the two-sided version.
Theorem 9.7. Let K be a rigid free bi-M -simplicial set with M -equivariant collapsing scheme
(E, R, C, c, i). Then, with the above notation, there is a free bi-M -CW complex Y with Y ≃M
|K| and such that the cells of Y are in bijective correspondence with E, and under this bijective
correspondence Yn is a free bi-M -set with basis EnB for all n.
Proof. This follows by applying Theorem 9.6 to the rigid free left M ×M op -simplicial set K. 
10. Guarded collapsing schemes
In this section we introduce the idea of a left guarded collapsing scheme. We shall prove that
−−→
whenever BM admits a left guarded collapsing scheme then EM will admit an M -equivariant
collapsing scheme whose M -orbits of cells are in bijective correspondence with the essential

46

ROBERT D. GRAY AND BENJAMIN STEINBERG

cells of the given collapsing scheme for BM . Applying Theorem 9.6 it will then follow that
when BM admits a left guarded collapsing scheme of finite type then the monoid M is of
type left-F∞ . Analogous results will hold for right guarded and right-F∞ , and (two-sided)
guarded and bi-F∞ . In later sections we shall give some examples of monoids which admit
guarded collapsing schemes of finite type, including monoids with finite complete presentations
(rewriting systems), and factorable monoids in the sense of [HO14].
S
Definition 10.1 (Guarded collapsing schemes). Let K = i≥0 Ki be a simplicial set and let
X be its geometric realisation. We identify the cells of X with the non-degenerate simplices of
K, and suppose that K admits a collapsing scheme (E, C, R, c, i). We say that this collapsing
scheme is
• left guarded if for every redundant n-cell τ we have i(τ ) 6= 0;
• right guarded if for every redundant n-cell τ we have i(τ ) 6= n + 1;
• guarded if it is both left and right guarded.
In other words, the collapsing scheme is guarded provided the function i never takes either
of its two possible extreme allowable values. The aim of this section is to prove the following
result.
Theorem 10.2. Let M be a monoid and suppose that BM admits a collapsing scheme (E, C, R, c, i).
(a) If (E, C, R, c, i) is left guarded , then there is an M -equivariant collapsing scheme (E, R, C, κ, ι)
−−→
for the free left M -simplicial set EM such that, for each n, the set of essential n-cells En
is a free left M -set of rank |En |.
(b) If (E, C, R, c, i) is right guarded , then there is an M -equivariant collapsing scheme (E, R, C, κ, ι)
←−−
for the free right M -simplicial set EM such that, for each n, the set of essential n-cells En
is a free right M -set of rank |En |.
(c) If (E, C, R, c, i) is guarded , then there is an M -equivariant collapsing scheme (E, R, C, κ, ι)
←−→
for the free bi-M -simplicial set EM such that, for each n, the set of essential n-cells En is
a free M × M op -set of rank |En |.
Corollary 10.3. Let M be a monoid and let |BM | be its classifying space.
(a) If BM admits a left guarded collapsing scheme of finite type then M is of type left F∞ (and
therefore also is of type left-FP∞ ).
(b) If BM admits a right guarded collapsing scheme of finite type then M is of type right-F∞
(and therefore also is of type right-FP∞ ).
(c) If BM admits a guarded collapsing scheme of finite type then M is of type bi-F∞ (and
therefore also is of type bi-FP∞ ).
Proof. This follows directly from Theorem 9.6 and its dual, and Theorems 9.7 and 10.2.



We shall give some examples of monoids to which this corollary applies in the next section.
The rest of this section will be devoted to the proof of Theorem 10.2. Clearly part (b) of the
theorem is dual to part (a). The proofs of parts (a) and (c) are very similar, the only difference
being that the stronger guarded condiion is needed for (c), while only left guarded is needed
for (a). We will begin by giving full details of the proof of Theorem 10.2(a). Then afterwards
we will explain the few modifications in the proof necessary to obtain the two-sided proof for
(c), in particular highlighting the place where the guarded condition is needed.
10.1. Proof of Theorem 10.2(a). Let (E, C, R, c, i) be a left guarded collapsing scheme for
BM . We can now ‘lift’ this collapsing scheme to a collapsing scheme (E, R, C, κ, ι) for the

TOPOLOGICAL FINITENESS PROPERTIES

47

−−→
simplicial set EM in the following natural way. First observe that
m(m1 , m2 , ..., mn ) is an n-cell of EM
⇔ mi 6= 1 for all 1 ≤ i ≤ n
⇔ (m1 , m2 , ..., mn ) is an n-cell of BM .
−−→
Define an n-cell m(m1 , m2 , ..., mn ) of EM to be essential (respectively redundant, collapsible respectively) if and only if (m1 , m2 , ..., mn ) is essential (respectively redundant, collapsible
respectively) in the collapsing scheme (E, R, C, c, i). This defines the partition (E, R, C) of the
−−→
n-cells of EM for each n. We call these sets the essential, redundant and collapsible cells,
−−→
−−→
respectively, of EM . For the mappings κ and ι, given mτ ∈ EM where τ = (m1 , ..., mn ) is a
redundant n-cell of BM we define
ι(mτ ) = i(τ )
κ(mτ ) = m(c(τ )).

(10.1)
(10.2)

We claim that (E, R, C, κ, ι) is an M -equivariant collapsing scheme for the free left M -simplicial
−−→
set EM such that, for each n, the set of essential n-cells En is a free left M -set of rank |En |.
Once proved this will complete the proof of Theorem 10.2(a).
−−→
We begin by proving that (E, R, C, κ, ι) is a collapsing scheme for EM , and then we will
verify that all of the conditions (A1) to (A6) are satisfied.
10.1.1. Proving that (E, R, C, κ, ι) is a collapsing scheme.
Proposition 10.4. With the above definitions, (E, R, C, κ, ι) is a collapsing scheme for the
−−→
simplicial set EM .
−−→
We have already observed that (E, R, C) partitions the cells of EM . To complete the proof
of the proposition we need to verify that the conditions (C1) and (C2) in the definition of
collapsing scheme are both satisfied. For part of the proof we will find it useful to recast the
ideas in terms of certain bipartite digraphs. The idea of viewing collapsing schemes in terms
of matchings in bipartite graphs is a natural one and has been used in the literature; see for
example [Cha00]. Let us now introduce the terminology and basic observations about digraphs
that we shall need.
A directed graph D consists of: a set of edges ED, a set of vertices V D and functions α
and β from ED to V D. For e ∈ E we call α(e) and β(e) the initial and terminal vertices of
the directed edge e. A directed path of length n in D is a sequence of edges e1 e2 . . . en such
that β(ei ) = α(ei+1 ) for each directed edge. Note that edges in paths are allowed to repeat,
and vertices can also repeat in the sense that β(ei ) = β(ej ) is possible for distinct i and j
(in graph theory literature what we call a path here is often called a walk). By an infinite
directed path we mean a path (ei )i∈N . Note that an infinite directed path need not contain
infinitely many distinct edges. For example, if a digraph contains a directed circuit e1 e2 e3
then e1 e2 e3 e1 e2 e3 . . . would be an infinite directed path. A bipartite digraph D with bipartition
V D1 ∪ V D2 has vertex set V D = V D1 ∪ V D2 where V D1 and V D2 are disjoint, such that for
every e ∈ ED we have either α(e) ∈ V D1 and β(e) ∈ V D2 , or α(e) ∈ V D2 and β(e) ∈ V D1
(i.e., there are no directed edges between vertices in the same part of the bipartition). A
homomorphism ϕ : D → D ′ between digraphs is a map ϕ : (V D ∪ ED) → (V D ′ ∪ ED ′ ) which
maps vertices to vertices, edges to edges, and satisfies α(ϕ(e)) = ϕ(α(e)) and β(ϕ(e)) = ϕ(β(e)).
If p = e1 e2 . . . en is a path of length n in D and ϕ : D → D ′ is a digraph homomorphism then
we define ϕ(p) = ϕ(e1 )ϕ(e2 ) . . . ϕ(en ) which is a path of length n in D ′ . Note that in general a

48

ROBERT D. GRAY AND BENJAMIN STEINBERG

homomorphism is allowed to map two distinct vertices (resp. edges) of D to the same vertex
(resp. edge) of D ′ . Since digraph homomorphisms map paths to paths, we have the following
basic observation.
Lemma 10.5. Let ϕ : D → D ′ be a homomorphism of directed graphs. If D has an infinite
directed path than D ′ has an infinite directed path.
For each n ∈ N let Γ(n) (BM ) be the directed bipartite graph defined as follows. The vertex
set V Γ(n) (BM ) = Cn ∪ Cn+1 where Ci denotes the set of i-cells BM . The directed edges EV of
V are are of two types:
(DE1) A directed edge τ −→ dj (τ ) (with initial vertex τ and terminal vertex dj (τ )) for every
collapsible τ ∈ Cn+1 and j ∈ {0, . . . , n + 1} such that dj (τ ) is a redundant n-cell (i.e., is a
redundant non-degenerate n-simplex) and either c(dj (τ )) 6= τ or c(dj (τ )) = τ but j 6= i(dj (τ ));
(DE2) A directed edge σ −→ c(σ) (with initial vertex σ and terminal vertex c(σ)) for every
redundant σ ∈ Cn .
We sometimes refer to these two types of directed arcs as the “down-arcs” and the “uparcs” (respectively) in the bipartite graph. Note that condition (C2) in the collapsing scheme
definition is equivalent to saying that the digraph D(n, n + 1) does not contain any infinite
−−→
directed path, and in particular does not contain any directed cycles. Let Γ(n) (EM ) be the
corresponding directed bipartite graph defined in the same way using (E, R, C, κ, ι), with vertex
−−→
set the n and n + 1 cells of EM and directed edges determined by the maps κ and ι. To simplify
notation, let us fix n ∈ N and set Γ(BM ) = Γ(n) (BM ) and Γ(EM ) = Γ(n) (EM ).
−−→
−−→
Lemma 10.6. Let π : V Γ(EM ) ∪ EΓ(EM ) → V Γ(BM ) ∪ EΓ(BM ) be defined on vertices by:
m(m1 , . . . , mk ) 7→ (m1 , . . . , mk ) (k ∈ {n, n + 1})
and defined on edges (DE1) and (DE2) by π(x → y) = π(x) → π(y). Then π is a digraph
homomorphism.
−−→
Proof. We need to prove, for each directed edge x → y from EΓ(EM ), that π(x) → π(y) is a
directed edge in EΓ(BM ). There are two cases that need to be considered depending on arc
type. The two arc types depend on whether the arc is going downwards from level n + 1 to
level n (arc type 1) or upwards from level n to level n + 1 (arc type 2).
−−→
Case: Arc type 1. Let m(m1 , . . . , mn+1 ) be a collapsible n + 1 cell in EM and let j ∈
{0, . . . , n + 1} and suppose that
m(m1 , . . . , mn+1 ) −→ dj (m(m1 , . . . , mn+1 ))
−−→
−−→
is an arc in Γ(EM ). This means that dj (m(m1 , . . . , mn+1 )) is a redundant n-cell in EM and
if κ(dj (m(m1 , . . . , mn+1 ))) = m(m1 , . . . , mn+1 ), then j 6= ι(dj (m(m1 , . . . , mn+1 ))). Note that
j = 0 or j = n + 1 is possible here. We claim that
π(m(m1 , . . . , mn+1 )) −→ π(dj (m(m1 , . . . , mn+1 )))
−−→
in Γ(BM ). Indeed, we saw above in Section 5 that the projection π : EM → BM is a simplicial
morphism with
π(dj (m(m1 , . . . , mn+1 ))) = dj (m1 , . . . , mn+1 ) = dj (π(m(m1 , . . . , mn+1 ))).

TOPOLOGICAL FINITENESS PROPERTIES

49

Therefore
π(m(m1 , . . . , mn+1 ))
= (m1 , . . . , mn+1 ) → dj (m1 , . . . , mn+1 )
= π(dj (m(m1 , . . . , mn+1 ))),
in Γ(BM ), since if c(dj (m1 , . . . , mn+1 )) = (m1 , . . . , mn+1 ), then
κ(dj (m(m1 , . . . , mn+1 ))) = κ(mdj (m1 , . . . , mn+1 )) = mc(dj (m1 , . . . , mn+1 )) = m(m1 , . . . , mn+1 )
and hence by definition of ι we have
j 6= ι(dj (m(m1 , . . . , mn+1 ))) = i(dj (m1 , . . . , mn+1 )).
Case: Arc type 2. These are the arcs that go up from level n to level n + 1. A typical such
−−→
arc arises as follows. Let mσ ∈ Cn be a redundant cell in EM where σ is a redundant cell in
BM and m ∈ M . Then κ(mσ) ∈ Cn+1 is collapsible and
dι(mσ) (κ(mσ)) = mσ.
A typical type 2 arc has the form
mσ −→ κ(mσ) = m(c(σ)),
by definition of κ. Applying π to this arc gives
π(mσ) = σ −→ c(σ) = π(m(c(σ))),
which is a type 2 arc in Γ(BM ) completing the proof of the lemma.



Proof of Proposition 10.4. We must check that (E, R, C, κ, ι) satisfies the two collapsing scheme
conditions (C1) and (C2).
Verifying collapsing scheme condition (C1). We must prove that the map κ defines a
bijection from the redundant n-cells to the collapsible n + 1-cells.
The map is injective since κ(mτ ) = κ(m′ σ) implies mc(τ ) = m′ c(σ) so m = m′ and τ = σ
since c is injective by assumption. Also, given an arbitrary collapsible n + 1 cell mσ there exists
σ = c(τ ) where τ is a redundant n-cell an so mσ = κ(mτ ).
Moreover, for every redundant n-cell mτ , we have
dι(mτ ) (κ(mτ )) = di(τ ) (mc(τ )) (by definition)
= mdi(τ ) (c(τ )) (since i(τ ) 6= 0)
= mτ,
and this concludes the proof that collapsing scheme condition (C1) holds.
Note that it is in the second line of the above sequence of equations that we appeal to our
assumption that (E, C, R, c, i) is left guarded (which implies i(τ ) 6= 0). In fact, this will be the
only place in the proof of Theorem 10.2(a) that the left guarded assumption is used.
Verifying collapsing scheme condition (C2). To see that collapsing scheme condition
−−→
−−→
(C2) holds let Γ(BM ) and Γ(EM ) be the level (n, n + 1) bipartite digraphs of BM and EM ,
respectively, defined above. By Lemma 10.6 the mapping π defines a digraph homomorphism
−−→
−−→
from Γ(EM ) to Γ(BM ). If Γ(EM ) contained an infinite directed path then by Lemma 10.5
the image of this path would be an infinite directed path in Γ(BM ) which is impossible since

50

ROBERT D. GRAY AND BENJAMIN STEINBERG

−−→
(E, R, C, c, i) is a collapsing scheme. Therefore Γ(EM ) contains no infinite path and thus
(E, R, C, κ, ι) satisfies condition (C2).
This completes the proof of Proposition 10.4.

−−→
Remark 10.7. It follows from Proposition 10.4 that every down arc in Γ(EM ) (that is, every
arc of type (DE1)) is of the form τ → dj (τ ) where τ ∈ Cn+1 , dj (τ ) ∈ Rn , and κ(dj (τ )) 6= τ .
10.1.2. Proving (E, R, C, κ, ι) is a left M -equivariant collapsing scheme. To prove that (E, R, C, κ, ι)
−−→
is a left M -equivariant collapsing scheme for EM we need to verify that conditions (A1)–(A6)
−−→
hold. In Section 5 we proved that EM is a rigid free left M -simplicial set. In addition to
−−→
this, from the definitions an n-cell m(m1 , m2 , ..., mn ) of EM essential (redundant, collapsible
respectively) if and only if (m1 , m2 , ..., mn ) is essential (redundant, collapsible respectively) in
the collapsing scheme (E, R, C, c, i) of BM . These facts prove that (A1) and (A2) both hold:
−−→
(A1) The action of M on EM maps n-simplicies to n-simplicies, and commutes with di and
−−→
si , that is, M is acting by simplicial morphisms on EM .
(A2) For every n-simplex σ and m ∈ M , σ is a cell (i.e. is a non-degenerate simplex) if and
only if mσ is a cell, in which case σ ∈ E (respectively R, C) if and only if mσ ∈ E
(respectively R, C).
The next axiom we need to verify is:
(A3) If (σ, τ ) ∈ Rn × Cn+1 is a matched redundant-collapsible pair (i.e. τ = c(σ)) then so is
the pair m(σ, τ ) = (mσ, mτ ) ∈ Rn × Cn+1 .
−−→
We shall prove a little more than this. We consider the bipartite digraph Γ(EM ) between levels
n and n + 1 defined above. We want to prove that M acts on this digraph, that is, that arcs
are sent to arcs under the action. We note that the action of M on the vertices preserves the
bipartition. As above, there are two types of directed arcs that need to be considered, the
up-arcs and the down-arcs.
First consider the down-arcs. My Remark 10.7, a typical down-arc has the form
dj

mσ −→ dj (mσ)
where κ(dj (mσ)) 6= mσ. Let n ∈ M be arbitrary. Then we have
dj

nmσ −→ dj (nmσ) = ndj (mσ),
by definition of dj . This is a down-arc because if κ(dj (nmσ)) = nmσ, then from nmdj (σ) =
dj (nmσ) we deduce that nmc(dj (σ)) = nmσ and so c(dj (σ)) = σ, whence κ(dj (mσ)) =
κ(mdj (σ)) = mc(dj (σ)) = mσ, which is a contradiction.
Now consider up-arcs. A typical up-arc has the form mσ → κ(mσ). Let n ∈ M . Then
nmσ → κ(nmσ) = nm(c(σ)) = nκ(mσ),
which is an up-arc as required.
−−→
−−→
This covers all types of arcs in Γ(EM ) and we conclude that M acts on Γ(EM ) by digraph
endomorphisms. This fact together with (A2) then implies property (A3), since the up-arcs in
this bipartite graph are precisely the matched redundant-collapsible pairs. Next consider

TOPOLOGICAL FINITENESS PROPERTIES

51

(A4) There is a subset B ⊆ E ∪ R ∪ C such that for all n the set of n-cells is a free left M -set
with basis Bn (the n-cells in B). Let EB = E ∩ B, RB = R ∩ B and CB = C ∩ B. Then
En is a free left M -set with basis EB
n , and similarly for Rn and Cn .
−−→
We saw in Section 5 that the set of n-cells of EM is a free left M -set with basis the set of
n-cells
B = {(m1 , . . . , mn ) : mi 6= 1 for all i}
of BM . The last clause of (A4) then follows from (A2). Now we shall prove
(A5) For every matched pair (σ, τ ) ∈ R × C, σ ∈ RB if and only if τ ∈ CB . In particular, for
every matched pair (σ, τ ) there is a unique pair (σ ′ , τ ′ ) ∈ RB × CB and m ∈ M such
that (σ, τ ) = m(σ ′ , τ ′ ).
−−→
The matched pairs are the up-arcs in the graph Γ(EM ). A typical up-arc has the form
m(m1 , . . . , mn ) → κ(m(m1 , . . . , mn ) = mc(m1 , . . . , mn ).
So this pair is
m · ((m1 , . . . , mn ) → c(m1 , . . . , mn ))
where
(m1 , . . . , mn ) → c(m1 , . . . , mn )
is a uniquely determined matched basis pair. Also, if (σ, τ ) is a matched pair then σ =
(m1 , . . . , mn ) belongs to the basis if and only if κ(m1 , . . . , mn ) = c(m1 , . . . , mn ) belongs to
the basis, completing the proof of (A5). Finally we turn our attention to showing axiom
(A6) For every redundant cell τ and every m ∈ M
height(τ ) = height(mτ )
where height is taken with respect to the collapsing scheme (E, R, C, κ, ι).
The following lemma will be useful.
−−→
−−→
Lemma 10.8 (Path lifting property). Define a mapping π : V Γ(EM )∪EΓ(EM ) → V Γ(BM )∪
EΓ(BM ) defined on vertices by:
m(m1 , . . . , mk ) 7→ (m1 , . . . , mk ) (k ∈ {n, n + 1})
and defined on edges (DE1) and (DE2) by π(x → y) = π(x) → π(y). Let µ be a redundant
−−→
n-cell in EM . Then for each path p in Γ(BM ), with initial vertex π(µ) there is a path p̂ in
−−→
Γ(EM ), with initial vertex µ, such that π(p̂) = p.
Proof. We shall establish two claims, from which the lemma will be obvious by induction on
−−→
path length. First we claim that if y = mτ is a redundant n-cell of EM (with m ∈ M and τ and
n-cell of BM ) and σ ∈ V Γ(BM ) is a vertex such that there is a directed edge τ = π(y) → σ,
−−→
−−→
then there is a vertex z ∈ V Γ(EM ) such that y → z is a directed edge in EΓ(EM ), and
−−→
π(y → z) = π(y) → σ. Indeed, set z = κ(y). Then y → z is a directed edge in EΓ(EM ) by
definition and π(z) = π(κ(y)) = π(κ(mτ )) = π(m(c(τ ))) = c(τ ) = σ.
−−→
Next we claim that if x → y is an up-arc of EΓ(EM ) as in (DE2) and σ is a vertex in V Γ(BM )
−−→
such that π(y) → σ is a directed edge in EΓ(BM ), then there exists a vertex z ∈ V Γ(EM ) such

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ROBERT D. GRAY AND BENJAMIN STEINBERG

−−→
that y → z is a directed edge in EΓ(EM ), and π(y → z) = π(y) → σ. Write x = mτ where
m ∈ M and τ is a redundant n-cell of BM . Then x → y is equal to mτ −→ κ(mτ ) = mc(τ ). In
Γ(BM ) we have the path π(x) → π(y) → σ which equals τ → c(τ ) → σ. Therefore c(τ ) → σ is
an arc in Γ(BM ) of type 1. Therefore, by Remark 10.7 applied to the graph Γ(BM ), it follows
that σ is a redundant n-cell with σ = dj (c(τ )) for some j ∈ {0, . . . , n + 1} and σ 6= τ (using that
c is a bijection). Set z = dj (y). We need to show that π(z) = σ and that y → z is a directed
−−→
edge in Γ(EM ).
For the first claim, since π is a simplicial morphism we have
π(z) = π(dj (y)) = dj (π(y)) = dj (c(τ )) = σ.
−−→
For the second claim, to verify that y → z is a directed edge in Γ(EM ) we just need to
show that z is a redundant cell and y 6= κ(z). From the definitions it follows that under the
mapping π any vertex in the preimage of a redundant cell is a redundant cell. Thus, since σ is
redundant and π(z) = σ it follows that z is redundant. Finally, if y = κ(z), then z = x because
κ is injective. Therefore, σ = π(z) = π(x) = τ , which is a contradiction.
We can now construct the path p̂ by induction on the length of p, where the inductive step
uses the first claim if the lift of the previous portion ends at a redundant cell and uses the
second claim if it ends at a collapsible cell.

Axiom (A6) is then easily seen to be a consequence of the following lemma.
Lemma 10.9. Let τ be a redundant cell in the collapsing scheme (E, R, C, κ, ι). Write τ = mσ
−→ (mσ) denote the height of mσ with respect to
where σ is a redundant cell in BM . Let height−
EM
the collapsing scheme (E, R, C, κ, ι), and let heightBM (σ) denote the height of σ with respect to
−→ (mσ) = heightBM (σ).
the collapsing scheme (E, R, C, c, i). Then height−
EM
Proof. Let
mσ = τ = τ0 ≻ τ1 ≻ · · · ≻ τk
be a descending chain of redundant n-cells from R. It follows that there is a directed path
p = τ0 → κ(τ0 ) → · · · → κ(τk−1 ) → τk
−−→
in Γ(EM ). Since π is a digraph homomorphism it follows that π(p) is a directed path in Γ(BM )
and hence
σ = π(τ0 ) ≻ π(τ1 ) ≻ · · · ≻ π(τk )
−→ (mσ) ≤ heightBM (σ).
is a descending chain of redundant n-cells in R. This proves that height−
EM
For the converse, let

σ = σ0 ≻ σ1 ≻ · · · ≻ σk
be a descending chain of redundant n-cells from R. Then there is a directed path
q = σ0 → c(σ0 ) → · · · → c(σk−1 ) → σk
−−→
in Γ(BM ). By Lemma 10.8 we can lift q to a path q̂ in Γ(EM ) with initial vertex mσ and such
that π(q̂) = q. But then the redundant cells in the path q̂ form a decending chain of length k
−→ (mσ) ≥ heightBM (σ).

starting at mσ, proving that height−
EM
This completes the proof of Theorem 10.2(a) and its dual Theorem 10.2(b).

TOPOLOGICAL FINITENESS PROPERTIES

53

10.2. Proof of Theorem 10.2(c). We shall explain how the above proof of Theorem 10.2(c)
is modified to prove the two-sided analogue Theorem 10.2(c).
←−→
Let (E, C, R, c, i) be a guarded collapsing scheme. Define an n-cell m(m1 , m2 , ..., mn )u of EM
to be essential (respectively redundant, collapsible respectively) if and only if (m1 , m2 , ..., mn ) is
essential (respectively redundant, collapsible respectively) in the collapsing scheme (E, R, C, c, i).
This defines (E, R, C).
←−→
For the mappings κ and ι, given mτ u ∈ EM where τ = (m1 , ..., mn ) is an n-cell of BM we
define
ι(mτ u) = i(τ )
κ(mτ u) = m(c(τ ))u.

(10.3)
(10.4)

With these definitions we claim that (E, R, C, κ, ι) is an M -equivariant collapsing scheme for
←−→
the free bi-M -simplicial set EM such that for each n the set of essential n-cells En is a free
bi-M -set of rank |En |.
10.2.1. Proving that (E, R, C, κ, ι) is a collapsing scheme.
Proposition 10.10. With the above definitions, (E, R, C, κ, ι) is a collapsing scheme for the
←−→
simplicial set EM .
The proof is analogous to the proof of Proposition 10.4. As in the proof of that proposition,
←−→
←−→
we have a digraph homomorphism π : V Γ(EM ) ∪ EΓ(EM ) → V Γ(BM ) → EΓ(BM ) given by
m(m1 , . . . , mk )n → (m1 , . . . , mk ),
π(x → y) = π(x) → π(y).
To verify collapsing scheme condition (C1) it suffices to observe that for every redundant n-cell
mτ we have
dι(mτ n) (κ(mτ n)) = di(τ ) (mc(τ )n) (by definition)
= mdi(τ ) (c(τ ))n (since i(τ ) 6= 0 and i(τ ) 6= n + 1)
= mτ n.
Here the second line appeals to the fact that the original collapsing scheme is guarded. Collapsing scheme condition (C2) holds again by applying Lemma 10.5 and the fact that π is a
digraph homomorphism.
10.2.2. Proving (E, R, C, κ, ι) is an M -equivariant collapsing scheme. We have already seen in
←−→
Section 5 that EM is a bi-M -simplicial set. We need to show that (E, R, C, κ, ι) is an M ←−→
equivariant collapsing scheme for EM . By definition, for this we need to verify that axioms
←−→
(A1)-(A6) are satisfied for EM as a left M × M op -simplicial set.
←−→
In Section 5 we saw that EM is a rigid free left M × M op -simplicial set. Together with the
←−→
definition of the action of M × M op on EM and the definition of E, R and C, axioms (A1) and
(A2) both then follow. Axioms (A3)-(A6) are then proved exactly as above just with M × M op
in place of M in the proof. This then completes the proof of Theorem 10.2(c).
11. Monoids admitting guarded collapsing schemes
In this section we give examples of classes of monoids to which the above theory of equivariant
collapsing schemes applies. In particular, this will allow us to use the theory developed above

54

ROBERT D. GRAY AND BENJAMIN STEINBERG

to give a topological proof of the fact that monoids which admit finite complete presentations
are of type bi-F∞ .
Let M be a monoid defined by a finite presentation hA | Ri with generators A and defining
∗ is the smallest congruence
∗ where ↔
relations R ⊆ A∗ × A∗ . Thus, M is isomorphic to A∗ / ↔
R
R
∗
on A containing R. We view hA | Ri as a string rewriting system, writing l → r for the pair
(l, r) ∈ R. We define a binary relation → on A∗ , called a single-step reduction, in the following
way:
u → v ⇔ u ≡ w1 lw2 and v ≡ w1 rw2
for some (l, r) ∈ R and w1 , w2 ∈ X ∗ . A word is called irreducible if no single-step reduction
∗ .
rule may be applied to it. The transitive and reflexive closure of →R is denoted by −→
R
This rewriting system is called noethereian if there are no infinite descending chains
w1 →R w2 →R w3 →R · · · →R wn →R · · · .
∗ v ′ there is a word w ∈ X ∗ such
∗ v and u −→
It is called confluent if whenever we have u −→
R
R
∗ w. If R is simultaneously noetherian and confluent we say that R is
∗ w and v ′ −→
that v −→
R
R
complete. The presentation hA | Ri is called complete if the rewriting system R is complete.
It is well-known (see for example [BO93]) that if hA | Ri is a finite complete presentation then
∗ -class of A∗ contains exactly one irreducible element. So the set of irreducible elements
each ↔
R
give a set of normal forms for the elements of the monoid M . In particular, if a monoid admits
a presentation by a finite compete rewriting system then the word problem for the monoid is
decidable.
In [Bro92, page 145] a method is given for constructing a collapsing scheme on BM for any
monoid M that is given by a finite complete rewriting system. It is easily observed from [Bro92,
page 145] that in the given collapsing scheme (E, R, C, c, i) the function i never takes either of its
two extreme allowable values, that is, the collapsing scheme for BM given in [Bro92] is guarded
in the sense of Definition 10.1. Also, as Brown observes (see [Bro92, page 147]), it follows easily
from his definition that there are only finitely many essential cells in each dimension. Thus we
have:
Proposition 11.1. Let M be a monoid. If M admits a presentation by a finite complete
rewriting system then BM admits a guarded collapsing scheme of finite type.
It follows that the theory of M -equivariant collapsing schemes developed in the previous
section applies to monoids with complete presentations, giving:
Corollary 11.2. Let M be a monoid that admits a presentation by a finite complete rewriting
system. Then M is of type left-F∞ , right-F∞ and bi-F∞ .
Proof. By Proposition 11.1 the simplicial set BM admits a guarded collapsing scheme of finite
type. Then the result follows from Corollary 10.3.

We obtain the following result of Kobayashi as a special case.
Corollary 11.3 ([Kob05]). Let M be a monoid that admits a presentation by a finite complete
rewriting system. Then M is of type bi-FP∞ .
Proof. Follows from Proposition 7.7 and Corollary 11.2.



More recently the class of, so-called, factorable monoids was introduced in work of Hess
and Ozornova [HO14]. Since it is quite technical we shall not give the definition of factorable
monoid here, we refer the reader to [HO14] to the definition, and we shall use the same notation
as there. In their work they show that a number of interesting classes of monoids admit

TOPOLOGICAL FINITENESS PROPERTIES

55

factorability structures. In some cases (e.g. Garside groups) the rewriting systems associated
with factorability structures are finite and complete, and so in these cases the monoids are
bi-F∞ . On the other hand, in [HO14, Appendix] they give an example of a factorable monoid
where the associated rewriting system is not noetherian and thus not complete (although it is
not discussed there whether the monoid admits a presentation by some other finite complete
presentation). So, there are some examples where factorability structures may be seen to exist,
even when the given presentation is not complete. In [HO14, Section 9] the authors construct
a matching on the reduced, inhomogeneous bar complex of a factorable monoid. As they say
in their paper, the matching they construct is very similar to the construction used by Brown
giving a collapsing scheme for monoids defined by finite complete rewriting systems [Bro92, page
145]. Details of the matching they construct for factorable monoids may be found on pages 27
and 28 of [HO14]. It is immediate from the definition of the mapping µ on page 28 that their
construction defines a guarded collapsing scheme for the simplicial set BM where M is a any
factorable monoid (M, E, η). If the generating set E for the monoid is finite, then the number of
essential cells in each dimension is finite, and so we have a guarded collapsing scheme of finite
type, giving:
Corollary 11.4. Let M be a monoid. If M admits a factorability structure (M, E, η) with finite
generating set E then BM admits a guarded collapsing scheme of finite type. In particular M
is of type left-F∞ , right-F∞ and bi-F∞ .
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Comput. Sci., 131, no. 2 (1994), pp. 271–294. doi: 10.1016/0304-3975(94)90175-9.

TOPOLOGICAL FINITENESS PROPERTIES

[Squ87]
[Sri96]
[Sta68]
[Ste92]
[Swa69]
[Wal65]
[Wei94]
[Wei13]

59

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Society, Providence, RI, 2013. An introduction to algebraic K-theory.

School of Mathematics,, University of East Anglia,, Norwich NR4 7TJ,, England and Department of Mathematics,, City College of New York,, Convent Avenue at 138th Street,, New York,
New York 10031,, USA
E-mail address: Robert.D.Gray@uea.ac.uk and bsteinberg@ccny.cuny.edu

