A NORMAL GENERATING SET FOR THE TORELLI GROUP OF A
COMPACT NON-ORIENTABLE SURFACE

Abstract. For a compact surface S, let I(S) denote the Torelli group of S. For a
compact orientable surface Σ, I(Σ) is generated by BSCC maps and BP maps (see [10]
and [11]). For a non-orientable closed surface N , I(N ) is generated by BSCC maps and
BP maps (see [5]). In this paper, we give an explicit normal generating set for I(Ngb ),
where Ngb is a genus-g compact non-orientable surface with b boundary components for
g ≥ 4 and b ≥ 1.

1. Introduction
For g ≥ 1 and b ≥ 0, let Ngb denote a genus-g compact connected non-orientable surface
with b boundary components, and let Ng = Ng0 . In this paper, we regard Ngb as a surface
obtained by attaching g Möbius bands to a sphere with g + b boundary components, as
shown in Figure 1. We say each of these Möbius bands attached to this sphere a cross cap.
The mapping class group M(Ngb ) of Ngb is the group consisting of isotopy classes of all
diffeomorphisms over Ngb which fix each point of the boundary. The Torelli group I(Ngb )
of Ngb is the subgroup of M(Ngb ) consisting of elements acting trivially on the integral
first homology group H1 (Ngb ; Z) of Ngb . The Torelli group of a compact orientable surface
is generated by BSCC maps and BP maps (see [10] and [11]). In particular, Johnson
[6] showed that the Torelli group of an orientable closed surface is finitely generated by
BP maps. Hirose and the author [5] showed that I(Ng ) is generated by BSCC maps and
BP maps for g ≥ 4. In this paper, we give an explicit normal generating set for I(Ngb )
consisting of BSCC maps and BP maps, for g ≥ 4 and b ≥ 1.

attach

{
{

arXiv:1601.06257v2 [math.GT] 14 Feb 2017

RYOMA KOBAYASHI

g

b

Figure 1. A genus-g compact connected non-orientable surface with b
boundary components.
Let N be a compact connected non-orientable surface. For a simple closed curve c on
N, we call c an A-circle (resp. an M-circle) if its regular neighborhood is an annulus
(resp. a Möbius band), as shown in Figure 2. For an A-circle c, we can define the mapping
class tc , called the Dehn twist about c, and the direction of the twist is indicated by a
1

2

R. KOBAYASHI

(a) A-circles.

(b) M-circles.

Figure 2.

c

tc

Figure 3. The Dehn twist tc about c.
small arrow written beside c as shown in Figure 3. We can notice that an A-circle (resp.
an M-circle) passes through cross caps even times (resp. odd times).
Let α, β, β ′ , γ, δi , ρi , σij and σ̄ij be simple closed curves on Ngb as shown in Figure 4.
The main result of this paper is as follows.
Theorem 1.1. Let g ≥ 5 and b ≥ 0. In M(Ngb ), I(Ngb ) is normally generated by tα ,
b
b
tβ t−1
β ′ , tδi , tρi , tσij and tσ̄ij for 1 ≤ i, j ≤ b − 1 with i < j. In M(N4 ), I(N4 ) is normally
generated by tα , tβ t−1
β ′ , tδi , tρi , tσij , tσ̄ij and tγ for 1 ≤ i, j ≤ b − 1 with i < j.

β
α

σij

'

δi

β

i

j

(a) Loops α, β, β ′ , δi and σ̄ij .

σij

ρi
γ
i

j

(b) Loops γ, ρi and σij .

Figure 4.
In this paper, for f, g ∈ M(Ngb ), the composition gf means that we first apply f and
then g.
2. Basics on mapping class groups for non-orientable surfaces
2.1. On mapping class groups for non-orientable surfaces.
Mapping class groups for orientable surfaces are generated by only Dehn twists. Lickorish showed that M(Ng ) is generated by Dehn twists and Y -homeomorphisms and that
the subgroup of M(Ng ) generated by all Dehn twists is an index 2 subgroup of M(Ng )
(see [7, 8]). Hence M(Ng ) is not generated by only Dehn twists. On the other hand,

A NORMAL GENERATING SET FOR I(Ngb )

3

since a Y -homeomorphism acts trivially on H1 (Ng ; Z/2Z), M(Ng ) is not generated by
only Y -homeomorphisms.
Chillingworth [2] found a finite generating set for M(Ng ). M(N1 ) and M(N11 ) are
trivial (see [3]). Finite presentations for M(N2 ), M(N21 ), M(N3 ) and M(N4 ) are obtained
by [7], [12], [1] and [14], respectively. Paris-Szepietowski [9] obtained a finite presentation
of M(Ngb ) for b = 0, 1 and g + b > 3.
Let N be a non-orientable surface, and let a and m be an oriented A-circle and an
M-circle on N respectively such that a and m mutually intersect transversely at only one
point. We now define a Y -homeomorphism Ym,a . Let K be a regular neighborhood of a∪b
in N, and let M be a regular neighborhood of m in the interior of K. We can see that
K is homeomorphic to the Klein bottle with one boundary component. Ym,a is defined as
the isotopy class of a diffeomorphism over N which is described by pushing M once along
a and which fixes each point of the boundary and the exterior of K (see Figure 5).

K

a

m
Ym,a
M
Figure 5. The Y -homoemorphism Ym,a .
2.2. On Torelli groups for non-orientable surfaces.
Let c be an A-circle on a non-orientable surface N such that N \ c is not connected.
We call tc a bounding simple closed curve map, for short a BSCC map. For example, in
Theorem 1.1, tα , tγ , tδi , tρi and tσij are BSCC maps. Let c1 and c2 be A-circles on N such
that N \ci is connected, N \(c1 ∪c2 ) is not connected and one of its connected components
is an orientable surface with two boundary components. We call tc1 t−1
c2 a bounding pair
−1
map, for short a BP map. For example, in Theorem 1.1, tβ tβ ′ is a BP map.
Hirose and the author [5] obtained the following theorem.
Theorem 2.1 ([5]). For g ≥ 5, I(Ng ) is normally generated by tα and tβ t−1
β ′ in M(Ng ).
−1
I(N4 ) is normally generated by tα , tβ tβ ′ and tγ in M(N4 ).
Theorem 1.1 is a natural extension of Theorem 2.1.
We can check that all BSCC maps and BP maps are in I(Ngb ). In addition, by Theorem 1.1, we have that I(Ngb ) is generated by BSCC maps and BP maps. We do not know
whether or not I(Ngb ) can be finitely generated.
2.3. Capping, Pushing and Forgetful homomorphisms.
Let N be a compact non-orientable surface. Take a point ∗ in the interior of N. Let
M(N, ∗) denote the group consisting of isotopy classes of all diffeomorphisms over N
which fix ∗ and each point of the boundary, and let I(N, ∗) denote the subgroup of
M(N, ∗) consisting of elements acting trivially on H1 (N; Z)
Take a point ∗ in the interior of Ngb−1 . We regard Ngb as a subsurface of Ngb−1
not containing ∗. The natural embedding Ngb ֒→ Ngb−1 induces the homomorphism
Cgb : M(Ngb ) → M(Ngb−1 , ∗), called the capping homomorphism. We have the following
lemma.
Lemma 2.2 (cf. [4, 13]). ker Cgb is generated by tδb .

4

R. KOBAYASHI

Since tδb is in I(Ngb ) we obtain the following.
Corollary 2.3. ker Cgb |I(Ngb ) is generated by tδb .
We remark that Cgb and Cgb |I(Ngb ) are not surjective.
The pushing homomorphism Pgb−1 : π1 (Ngb−1 , ∗) → M(Ngb−1 , ∗) is defined as follows.
For x ∈ π1 (Ngb−1 , ∗) take a representative oriented loop x̃ based at ∗. Pgb−1 (x) is described
by pushing ∗ once along x̃ and fixes each point of exterior of neighborhood of x̃. Note
that Pgb−1 is an anti-homomorphism, that is, for x, y ∈ π1 (Ngb−1 , ∗) we have Pgb−1 (xy) =
Pgb−1 (y)Pgb−1 (x). The forgetful homomorphism Fgb−1 : M(Ngb−1 , ∗) → M(Ngb−1 ) is defined
naturally. Note that Fgb−1 is surjective.
We have the natural exact sequence
Fgb−1

Pgb−1

π1 (Ngb−1 , ∗) −→ M(Ngb−1 , ∗) −→ M(Ngb−1 ) −→ 1.
Since ImPgb−1 is in I(Ngb−1 , ∗) and Fgb−1 (I(Ngb−1 , ∗)) is equal to I(Ngb−1 ), we have the
exact sequence
π1 (Ngb−1 , ∗) −→ I(Ngb−1 , ∗) −→ I(Ngb−1 ) −→ 1.
3. A normal generating set fot Cgb (I(Ngb )) in Cgb (M(Ngb ))
By the capping homomorphism Cgb : M(Ngb ) → M(Ngb−1 , ∗) we have that a normal
generating set for I(Ngb ) in M(Ngb ) consists of tδb and lifts by Cgb of normal generators of
Cgb (I(Ngb )) in Cgb (M(Ngb )). Thus in this section we consider a normal generating set for
Cgb (I(Ngb )) in Cgb (M(Ngb )).
Let αi and βj be oriented loops on Ngb−1 based at ∗ as shown in Figure 6, and let xi
and yj be the elements of π1 (Ngb−1 , ∗) corresponding to αi and βj respectively. Note that
π1 (Ngb−1 , ∗) is generated by xi and yj for 1 ≤ i ≤ g and 1 ≤ j ≤ b − 1.

*
αg

β1

βb-1

{
{

α1
g

b-1

Figure 6. The oriented loops αi and βj on Ngb−1 .
Let p : π1 (Ngb−1 , ∗) → π1 (Ng , ∗) denote the natural surjection defined as p(xi ) = xi and
p(yj ) = 1, for b ≥ 1. For x ∈ π1 (Ngb−1 , ∗) we can denote p(x) = xεi11 xεi22 · · · xεitt , where t is
the word length of p(x) and εk = ±1. We define
Oi (x) = ♯{i2k−1 | i2k−1 = i},
Ei (x) = ♯{i2k | i2k = i},
Γb−1
= {x ∈ π1 (Ngb−1 , ∗) | Oi (x) = Ei (x), 1 ≤ i ≤ g}.
g
−2
−1 3 −1
For example, for x = x1 y2 x2 x−1
∈ π1 (Ngb−1 , ∗), since p(x) =
3 y5 y1 x1 x2 y4 x3
−1
−1 −1
x1 x2 x3 x1 x2 x3 , we have that Oi (x) and Ei (x) are equal to 1 (resp. 0) for i = 1, 2, 3
(resp. i ≥ 4), and hence x is in Γgb−1 .

A NORMAL GENERATING SET FOR I(Ngb )

5

In this section we prove the following three propositions.
Proposition 3.1. Im Fgb−1 |Cgb (I(Ngb )) is equal to I(Ngb−1 ).
Proposition 3.2. ker Fgb−1 |Cgb (I(Ngb )) is equal to Pgb−1 (Γb−1
g ).
b−1 2
b−1
Proposition 3.3. Pgb−1 (Γb−1
g ) is the normal closure of Pg (xg ), Pg (yj ) and
b
b
Pgb−1 (xg yj x−1
g ) for 1 ≤ j ≤ b − 1 in Cg (M(Ng )).

By Proposition 3.1 and Proposition 3.2, we have the exact sequence
Γb−1
−→ Cgb (I(Ngb )) −→ I(Ngb−1 ) −→ 1.
g
Hence Cgb (I(Ngb )) is the normal closure of Pgb−1 (Γgb−1 ) and lifts by Fgb−1 |Cgb (I(Ngb )) of normal
generators of I(Ngb−1 ). In addition, by Proposition 3.3 we obtain the following.
Corollary 3.4. In Cgb (M(Ngb )), Cgb (I(Ngb )) is normally generated by Pgb−1 (x2g ), Pgb−1 (yj ),
b−1
Pgb−1 (xg yj x−1
|Cgb (I(Ngb )) of normal generators of I(Ngb−1 ), for 1 ≤ j ≤
g ) and lifts by Fg
b − 1.
3.1. Proof of Proposition 3.1.
To prove Proposition 3.1, it suffices to show that for any ϕ ∈ I(Ngb−1 ) there is ϕ
e ∈ I(Ngb )
such that (Fgb−1 ◦ Cgb )(ϕ)
e = ϕ.
Let γi and δj be oriented loops on Ngb as shown in Figure 7, and let ci and di be the
elements of H1 (Ngb ; Z) corresponding to γi and δj respectively. As a Z-module, H1 (Ngb , Z)
has a presentation
H1 (Ngb , Z) = hc1 , . . . , cg , d1, . . . , db | 2(c1 + · · · + cg ) + (d1 + · · · + db ) = 0i,
and is isomorphic to Zg+b−1 .

γg

δ1

δb

{
{

γ1
g

b

Figure 7. The oriented loops γi and δj on Ngb .
For f ∈ M(Ngb ) we denote by f∗ the automorphism over H1 (Ngb ; Z) induced by f . Since
f fixes each point of the boundary of Ngb , we have that f∗ (dj ) = dj for 1 ≤ j ≤ b.
For any ϕ ∈ I(Ngb−1 ) there exists ψ ∈ M(Ngb ) such that (Fgb−1 ◦ Cgb )(ψ) = ϕ. For
1 ≤ i ≤ g there is an integer ni such that ψ∗ (ci ) = ci + ni db . Let γij and e
γij be simple
b
closed curves on Ng as shown in Figure 8, and let τij = tγij tγeij , for 1 ≤ i < j ≤ g. τij is
a mapping class which is described by pushing the b-th boundary component once along
between γij and e
γij . We can check

 ci − db (t = i),
cj + db (t = j),
(τij )∗ (ct ) =

ct
(t 6= i, j).

6

R. KOBAYASHI

γij
~
γ
ij
i

j

Figure 8. The loops γij and e
γij .

n

g−1
n2 n1
Let τ = τg−1g
· · · τ2g
τ1g and ϕ
e = τ ψ. Since (Fgb−1 ◦ Cgb )(τ ) = 1 we have (Fgb−1 ◦ Cgb )(ϕ)
e =
ϕ. For 1 ≤ i ≤ g − 1 we calculate

ϕ
e∗ (ci ) =
=
=
=
=
=

τ∗ (ψ∗ (ci ))
τ∗ (ci + ni db )
τ∗ (ci ) + ni τ∗ (db )
(τig )n∗ i (ci ) + ni db
(ci − ni db ) + ni db
ci .

In addition, we calculate
ϕ
e∗ (cg ) = τ∗ (ψ∗ (cg ))
= τ∗ (cg + ng db )
= τ∗ (cg ) + ng τ∗ (db )
= (cg +

g−1
X

nk db ) + ng db

k=1

= cg +

g
X

nk db .

k=1

Let c = 2(c1 + · · · + cg ) + (d1 + · · · + db )(= 0). We see
ϕ
e∗ (c) = 2(ϕ
e∗ (c1 ) + · · · + ϕ
e∗ (cg )) + (ϕ
e∗ (d1 ) + · · · + ϕ
e∗ (db ))
g
X
nk db ) + (d1 + · · · + db )
= 2(c1 + · · · + cg−1 + cg +
k=1

g

= c+2

X

nk db

k=1

= 2

g
X

nk db .

k=1

Since ϕ
e∗ (c) = 0 we have

g
X
k=1

nk = 0, and hence ϕ
e∗ (cg ) = cg . Hence we have that ϕ
e∗ is

the identity. Therefore we conclude that ϕ
e is in I(Ngb ) with (Fgb−1 ◦ Cgb )(ϕ)
e = ϕ. Thus we
complete the proof of Proposition 3.1.
3.2. Proof of Proposition 3.2.

A NORMAL GENERATING SET FOR I(Ngb )

7

Note that Pgb−1 (x) can be lifted by Cgb if and only if the word length of p(x) ∈ π1 (Ng , ∗)
is even. Let π1+ (Ngb−1 , ∗) denote the subgroup of π1 (Ngb−1 , ∗) consisting of x such that the
word length of p(x) is even. For x ∈ π1+ (Ngb−1 , ∗), let x∗ denote the automorphism over
H1 (Ngb , Z) induced by the natural lift by Cgb of Pgb−1 (x). For example, the natural lifts of
Pgb−1 (x2g ) and Pgb−1 (yj ) are tρb and tσjb t−1
δj , respectively.
−1
2
We can see that (xi )∗ = 1 and (yj )∗ = 1. Hence we have that (xi xj )∗ , (xi x−1
j )∗ , (xi xj )∗ ,
−1
−1
(x−1
i xj )∗ and (xj xi )∗ are mutually equal. Let xij = (xi xj )∗ for 1 ≤ i, j ≤ g. Note that

(γij )∗ (i < j),
xij =
−1
(γji
)∗ (j < i).
Lemma 3.5. For 1 ≤ i, j, k, l ≤ g, we have xij xkl = xkl xij .
Proof. If (i, j, k, l) = (i, i, i, i), (i, i, k, i), (i, i, i, l), (i, i, k, k), (i, j, i, j), (i, j, j, i) and
(i, i, k, l), it is clear that xij xkl = xkl xij . Hence we check the cases where (i, j, k, l) =
(i, j, i, l), (i, j, k, j), (i, j, k, i) and (i, j, k, l). For any 1 ≤ i, j ≤ g since xij acts trivially on
d1 , d2 , . . . , db , we check the actions on c1 , c2 , . . . , cg .
For any mutually different indices 1 ≤ i, j, l ≤ g we see

x (c − db ) = ci − 2db (t = i),


 ij i
xij (cj )
= cj + db (t = j),
xij xil (ct ) =
xij (cl + db ) = cl + db (t = l),


 x (c )
= ct
(t 6= i, j, l),
ij t

x (c − db ) = ci − 2db (t = i),


 il i
xil (cj + db ) = cj + db (t = j),
xil xij (ct ) =
xil (cl )
= cl + db (t = l),


 x (c )
= c
(t 6= i, j, l).
il

t

t

Hence we have xij xil = xil xij .
For any mutually different indices 1 ≤ i, j, k ≤ g we see

x (c )
= ci − d b


 ij i
xij (cj + db ) = cj + 2db
xij xkj (ct ) =
xij (ck − db ) = ck − db


 x (c )
= ct
ij t

x (c − db ) = ci − db


 kj i
xkj (cj + db ) = cj + 2db
xkj xij (ct ) =
xkj (ck )
= ck − d b


 x (c )
= c
kj

t

Hence we have xij xkj = xkj xij .
For any mutually different indices 1 ≤ i, j, k

x (c + db )


 ij i
xij (cj )
xij xki (ct ) =
xij (ck − db )


 x (c )
ij t

x (c − db )


 ki i
xki (cj + db )
xki xij (ct ) =
xki (ck )


 x (c )
ki t

t

(t = i),
(t = j),
(t = k),
(t 6= i, j, k),
(t = i),
(t = j),
(t = k),
(t 6= i, j, k).

≤ g we see
=
=
=
=

ci
cj + d b
ck − d b
ct

(t = i),
(t = j),
(t = k),
(t 6= i, j, k),

=
=
=
=

ci
cj + d b
ck − d b
ct

(t = i),
(t = j),
(t = k),
(t 6= i, j, k).

8

R. KOBAYASHI

Hence we have xij xki = xki xij .
For any mutually different indices 1 ≤ i, j, k, l ≤ g we see

xij (ci )
= ci − db (t = i),




= cj + db (t = j),
 xij (cj )
xij (ck − db ) = ck − db (t = k),
xij xkl (ct ) =


xij (cl + db ) = cl + db (t = l),


 x (c )
= ct
(t 6= i, j, k, l),
ij t

xkl (ci − db ) = ci − db (t = i),




 xkl (cj + db ) = cj + db (t = j),
xkl (ck )
= ck − db (t = k),
xkl xij (ct ) =


xkl (cl )
= cl + db (t = l),


 x (c )
= ct
(t 6= i, j, k, l).
kl t

Hence we have xij xkl = xkl xij .
Thus we obtain the claim.



For x ∈ π1+ (Ngb−1 , ∗) denote p(x) = xεi11 xεi22 · · · xεi2l2l . Since (yj )∗ = 1, we have x∗ =
xi1 i2 xi3 i4 · · · xi2l−1 i2l . For 1 ≤ i ≤ g, let s = ♯{k | i2k−1 = i2k = i}, t = Oi (x) − s and
u = Ei (x) − s. Since xii = 1 and xkl (ci ) = ci , by Lemma 3.5 we have
x∗ (ci ) =
=
=
=
=

xk(u)i · · · xk(1)i · xij(t) · · · xij(1) · xsii (ci )
xk(u)i · · · xk(1)i · xij(t) · · · xij(1) (ci )
xk(u)i · · · xk(1)i (ci − tdb )
(ci + udb ) − tdb
ci + (u − t)db ,

using the suitable indices j(1), . . . , j(t) and k(1), . . . , k(u). Therefore we have that x ∈
Γb−1
if and only if x∗ = 1.
g
Note that ker Fgb−1 |Cgb (I(Ngb )) is equal to the intersection of ker Fgb−1 and Cgb (I(Ngb )). For
b−1
b
b
any x ∈ Γb−1
g , since x∗ = 1, we have that Pg (x) is in Cg (I(Ng )). In addition, since
Pgb−1 (x) is in ker Fgb−1 , we have that Pgb−1 (x) is in ker Fgb−1 |Cgb (I(Ngb )) . Hence we conclude
b−1
|Cgb (I(Ngb )) .
Pgb−1 (Γb−1
g ) ⊂ ker Fg
b−1
For any ϕ ∈ ker Fg |Cgb (I(Ngb )) , since ϕ is in ker Fgb−1 there exists x ∈ π1 (Ngb−1 , ∗) such
that ϕ = Pgb−1 (x). In addition, since ϕ is in Cgb (I(Ngb )) we have x∗ = 1, and hence x ∈ Γb−1
g .
b−1
b−1
b−1
b−1
b−1
Hence ϕ is in Pg (Γg ). Therefore we conclude ker Fg |Cgb (I(Ngb )) ⊂ Pg (Γg ).
Thus we complete the proof of Proposition 3.2.
3.3. Proof of Proposition 3.3.
We have the exact sequence
π1+ (Ngb−1 , ∗) −→ Cgb (M(Ngb )) −→ M(Ngb−1 ) −→ 1.
b
b
Hence a normal generator of Pgb−1 (Γb−1
g ) in Cg (M(Ng )) is the image of a normal generator
+
of Γb−1
in π1 (Ngb−1 , ∗). Therefore we consider about the normal generators of Γb−1
in
g
g
+
b−1
π1 (Ng , ∗).
Let πgb−1 denote the finitely presented group with the generators xi and yj , and with the
relators x2i , yj and [xi(1) xi(2) , xi(3) xi(4) ], where 1 ≤ i, i(1), i(2), i(3), i(4) ≤ g, 1 ≤ j ≤ b − 1
and [x, y] means xyx−1 y −1. There is the natural surjection ψ : π1 (Ngb−1 , ∗) → πgb−1 . We
first show the following lemma.

A NORMAL GENERATING SET FOR I(Ngb )

9

Lemma 3.6. Γb−1
is equal to ker ψ.
g
Proof. It is clear that Γb−1
⊃ ker ψ. We show Γb−1
⊂ ker ψ. Namely, it suffices to show
g
g
that x ≡ 1 modulo x2i , yj and [xi(1) xi(2) , xi(3) xi(4) ] for any x ∈ Γb−1
g .
b−1
For any x ∈ Γg we can denote x ≡ xi1 xi2 · · · xi2l modulo x2i and yj . Since
Oi1 (x) = Ei1 (x), there exists 1 ≤ t ≤ l such that i1 = i2t . Hence, modulo x2i , yj
and [xi(1) xi(2) , xi(3) xi(4) ], we calculate
x ≡
=
≡
=
≡
..
.
=
≡
≡

xi1 xi2 · · · xi2t · xi2t+1 · · · xi2l
[xi1 xi2 , xi3 xi4 ]xi3 xi4 xi1 xi2 · xi5 xi6 · · · xi2t · xi2t+1 · · · xi2l
xi3 xi4 xi1 xi2 · xi5 xi6 · · · xi2t · xi2t+1 · · · xi2l
xi3 xi4 [xi1 xi2 , xi5 xi6 ]xi5 xi6 xi1 xi2 · xi7 xi8 · · · xi2t · xi2t+1 · · · xi2l
xi3 xi4 xi5 xi6 xi1 xi2 · xi7 xi8 · · · xi2t · xi2t+1 · · · xi2l
xi3 · · · xi2t−2 [xi1 xi2 , xi2t−1 xi2t ]xi2t−1 xi2t xi1 xi2 · xi2t+1 · · · xi2l
xi3 · · · xi2t−1 · xi2t xi1 · xi2 · xi2t+1 · · · xi2l
xi3 · · · xi2t−1 · xi2 · xi2t+1 · · · xi2l .

Let x′ = xi3 · · · xi2t−1 · xi2 · xi2t+1 · · · xi2l . It immediately follows that the word length of
′
′
′
x′ is 2l − 2 and x′ is in Γb−1
g . Since Oi3 (x ) = Ei3 (x ), there exists 2 ≤ t ≤ l such that
i3 = i2t′ . Then, similarly there exists x′′ such that the word length of x′′ is 2l − 4, x′′ is in
Γgb−1 and x′ ≡ x′′ modulo x2i , yj and [xi(1) xi(2) , xi(3) xi(4) ]. Repeating the same operation,
we have that x ≡ x′ ≡ x′′ ≡ · · · ≡ 1 modulo x2i , yj and [xi(1) xi(2) , xi(3) xi(4) ]. Therefore we
have x ∈ ker ψ.
Thus we obtain the claim.

We next show the following lemma.
Lemma 3.7. πgb−1 has a presentation with
πgb−1 = hx1 , . . . , xg , y1 , . . . , yb−1 | x21 , . . . , x2g , y1 , . . . , yb−1, (xi xj xk )2 , 1 ≤ i < j < k ≤ gi.
Proof. It suffices to show that (xi xj xk )2 is a product of conjugations of some
[xi(1) xi(2) , xi(3) xi(4) ] and that [xi(1) xi(2) , xi(3) xi(4) ] is a product of conjugations of some
(xi xj xk )2 , modulo x21 , x22 , . . . , x2g .
For any 1 ≤ i < j < k ≤ g, modulo x21 , x22 , . . . , x2g we see
(xi xj xk )2 ≡ (xi xj xk )xj xj (xi xj xk )
= xi xj · xk xj · xj xi · xj xk
≡ xi xj · xk xj · (xi xj )−1 (xk xj )−1
= [xi xj , xk xj ].
If (i(1), i(2), i(3), i(4)) = (i, i, i, i), (i, i, i, j), (i, i, j, i), (i, j, i, i), (j, i, i, i)
(i, i, j, j), (i, j, i, j), (i, j, j, i), (i, i, j, k) and (i, j, k, k), we have immediately that
[xi(1) xi(2) , xi(3) xi(4) ] ≡ 1 modulo x21 , x22 , . . . , x2g . Hence we check the other cases.

10

R. KOBAYASHI

For mutually different indices 1 ≤ i, j, k, l ≤ g, modulo x21 , x22 , . . . , x2g we see
[xi xj , xi xk ] ≡
≡
[xi xj , xk xj ] ≡
≡
[xi xj , xk xi ] ≡
≡
[xi xj , xj xk ] ≡

xi xj xi xk xj xi xk xi
xi (xj xi xk )2 x−1
i ,
xi xj xk xj xj xi xj xk
(xi xj xk )2 ,
xi xj xk xi xj xi xi xk
(xi xj xk )2 ,
xi xj xj xk xj xi xk xj

≡
[xi xj , xk xl ] ≡
≡
≡

(xi xk xj )2 ,
xi xj xk xl xj xi xl xk
(xi xj xk )2 xk xj xi · xi xj xl (xl xj xi )2 xl xk
(xi xj xk )2 xk xl (xl xj xi )2 (xk xl )−1

For the relator (xi xj xk )2 , applying conjugations and taking their inverses, it suffices to
consider the case i < j < k. Thus we obtain the claim.

By Lemma 3.6 and Lemma 3.7 we have the short exact sequence
1 −→ Γb−1
−→ π1 (Ngb−1 , ∗) −→ πgb−1 −→ 1.
g
Let (πgb−1 )+ denote the quotient group of π1+ (Ngb−1 , ∗) by Γb−1
g . Then we have the short
exact sequence
1 −→ Γb−1
−→ π1+ (Ngb−1 , ∗) −→ (πgb−1 )+ −→ 1.
g
From presentations of π1+ (Ngb−1 , ∗) and (πgb−1 )+ , we obtain the normal generators of Γb−1
g
in π1+ (Ngb−1 , ∗).
−1
Lemma 3.8. π1+ (Ngb−1 , ∗) is the free group freely generated by xi x−1
g , xg xj , yk and xg yk xg
for 1 ≤ i ≤ g − 1, 1 ≤ j ≤ g and 1 ≤ k ≤ b − 1.

Proof. We use the Reidemeister Schreier method. π1+ (Ngb−1 , ∗) is an index 2 subgroup of
π1 (Ngb−1 , ∗). Let U = {1, xg }. Remark that U is a Schreier transversal for π1+ (Ngb−1 , ∗)
in π1 (Ngb−1 , ∗). Let X = {x1 , . . . , xg , y1, . . . , yb−1 }. For u ∈ U and x ∈ X, ux = 1 if
(u, x) = (1, yj ) or (xg , xi ) (resp. ux = xg if (u, x) = (1, xi ) or (xg , yj )). A generating set of
−1
π1+ (Ngb−1 , ∗) is defined as B = {uxux−1 | u ∈ U, x ∈ X, ux ∈
/ U}. We see 1xi 1xi = xi xg−1 ,
−1
xg xj xg xj −1 = xg xj , 1yk 1yk = yk , xg yk xg yk −1 = xg yk x−1
g , for 1 ≤ i ≤ g − 1, 1 ≤ j ≤ g
−1
−1
is not in B. Thus we
and 1 ≤ k ≤ b − 1. Since 1xg 1xg = xg x−1
g = 1 ∈ U, 1xg 1xg
obtain the claim.

−1
Lemma 3.9. (πgb−1 )+ has a presentation with the generators xi x−1
g , xg xj , yk and xg yk xg
for 1 ≤ i ≤ g − 1, 1 ≤ j ≤ g and 1 ≤ k ≤ b − 1, and the following relators
2
(1) xi x−1
g · xg xi , xg for 1 ≤ i ≤ g − 1,
(2) yj for 1 ≤ j ≤ b − 1,
−1
−1
(3) xi x−1
g · xg xj · xk xg · xg xi · xj xg · xg xk for 1 ≤ i < j < k ≤ g,
(4) xg xi · xi x−1
g for 1 ≤ i ≤ g − 1,
(5) xg yj x−1
for
1 ≤ j ≤ b − 1,
g
−1
−1
(6) xg xi · xj xg · xg xk · xi x−1
g · xg xj · xk xg for 1 ≤ i < j < k ≤ g.

A NORMAL GENERATING SET FOR I(Ngb )

11

Proof. We apply the Reidemeister Schreier method for the presentation of πgb−1 in
Lemma 3.7. By the argument similar to Lemma 3.8, (πgb−1 )+ is generated by xi x−1
g ,
−1
xg xj , yk and xg yk xg for 1 ≤ i ≤ g − 1, 1 ≤ j ≤ g and 1 ≤ k ≤ b − 1. Let R be
the set of the relators of πgb−1 in Lemma 3.7. A set of the relators of (πgb−1 )+ is defined as S = {uru−1 | u ∈ U, r ∈ R}, where U = {1, xg }. We see x2i = xi x−1
g · xg xi ,
2
−1
−1
−1
2 −1
yj = yj , (xi xj xk ) = xi xg · xg xj · xk xg · xg xi · xj xg · xg xk , xg xi xg = xg xi · xi x−1
g ,
−1
2 −1
−1
−1
−1
xg yj x−1
=
x
y
x
,
x
(x
x
x
)
x
=
x
x
·
x
x
·
x
x
·
x
x
·
x
x
·
x
x
.
Thus
we
g j g
g
i j k
g i
j g
g k
i g
g j
k g
g
g
obtain the claim.

By Lemma 3.8 and Lemma 3.9, Γb−1
is normally generated by the relators of (πgb−1 )+
g
b−1 2
of Lemma 3.9, in π1+ (Ngb−1 , ∗). Hence Pgb−1 (Γb−1
g ) is normally generated by Pg (xi ),
b−1
−1
b−1
2 −1
Pgb−1 (yj ), Pgb−1 ((xi xj xk )2 ), Pgb−1 (xg x2i x−1
g ), Pg (xg yj xg ) and Pg (xg (xi xj xk ) xg ).
2
2 −1
bound a Möbius
Representative loops of x2i , xg x2i x−1
g , (xi xj xk ) and xg (xi xj xk ) xg
b−1 2
b−1
2 −1
b−1
band (see Figure 9).
Therefore Pg (xi ), Pg (xg xi xg ), Pg ((xi xj xk )2 ) and
b−1
2 −1
Pg (xg (xi xj xk ) xg ) are conjugate to Pgb−1 (x2g ). Thus we complete the proof of Proposition 3.3.

*
x12

xg2

(a) Representative loops of x21 , x22 , . . . , x2g bound
a Möbius band. Similarly a representative loop
of xg x2i x−1
g bounds a Möbius band for 1 ≤ i ≤ g.

*
(x1x2x3)2
(b) A representative loop of (x1 x2 x3 )2 bounds a Möbius band. Similarly representative loops of (xi xj xk )2 and xg (xi xj xk )2 xg−1 bound
a Möbius band for 1 ≤ i < j < k ≤ g.

Figure 9.

4. Proof of Theorem 1.1
We have the following short exact sequence:
1 → ker Cgb |I(Ngb ) → I(Ngb ) → Cgb (I(Ngb )) → 1.
Hence I(Ngb ) is the normal closure of ker Cgb |I(Ngb ) and lifts by Cgb of normal generators of
Cgb (I(Ngb )). By Corollary 2.3 and Corollary 3.4, we have that I(Ngb ) is normally generated
−1
b
b−1
b
by tδb , the lift tρb by Cgb of Pgb−1 (x2g ), the lift tσjb t−1
δj by Cg of Pg (yj ), the lift tσ̄jb tδj by Cg
b−1
of Pgb−1 (xg yj x−1
◦ Cgb )|I(Ngb ) of normal generators of I(Ngb−1 ). We prove
g ) and lifts by (Fg

12

R. KOBAYASHI

Theorem 1.1 by induction on the number b of the boundary components of Ngb . We take
the natural lift by (Fgb−1 ◦ Cgb )|I(Ngb ) of a normal generator of I(Ngb−1 ).
At first, I(Ng ) is normally generated by tα , tβ t−1
β ′ and tγ (see [5]). Hence we have that
−1
1
I(Ng ) is normally generated by tα , tβ tβ ′ , tγ and tδ1 , tρ1 . Similarly we have that I(Ng2 ) is
normally generated by tα , tβ t−1
β ′ , tγ , tδ1 , tρ1 and tδ2 , tρ2 , tσ12 , tσ̄12 . For b ≥ 3, suppose that
b−1
I(Ng ) is normally generated by tα , tβ t−1
β ′ , tγ , tδi , tρi , tσij and tσ̄ij for 1 ≤ i, j ≤ b − 1
b
with i < j. Then we have that I(Ng ) is normally generated by tα , tβ t−1
β ′ , tγ , tδi , tρi , tσij ,
tσ̄ij and tδb , tρb , tσkb , tσ̄kb for 1 ≤ i, j, k ≤ b − 1 with i < j. Hence we obtain a normal
generating set for I(Ngb ). In particular, for g ≥ 5 since I(Ng ) is normally generated by tα
b
and tβ t−1
β ′ (see [5]), we do not need tγ as a normal generator of I(Ng ) for g ≥ 5. Finally,
proving the following lemma, we finish the proof of Theorem 1.1.
Lemma 4.1. tδb , tρb , tσkb and tσ̄kb are not needed as normal generators of I(Ngb ).
Proof. Let aij , bjk and ckl be simple closed curves on Ngb as shown in Figure 10, for
1 ≤ i, j ≤ g and 1 ≤ k, l ≤ b − 1. Let dm be a diffeomorphism defined by pushing the
m-th boundary component once along an arrow as shown in Figure 11. Remark that the
isotopy class of dm is not in M(Ngb ), since dm does not fix boundary. taij and tdm (aij )
are conjugate to tα . tbjk and tdm (bjk ) are conjugate to ether tρk or t−1
ρk . tckl and tdm (ckl )
are conjugate to tσkl , with m 6= k, l. tdk (ckl ) and tdl (ckl ) are conjugate to tσ̄−1
and tσ̄kl ,
kl
respectively. Hence it suffices to show that tδb , tρb , tσkb and tσ̄kb are products of these
Dehn twists.

i

j

aij

k

l
ckl

bjk
Figure 10. The loops aij , bjk and ckl .

1

g

1

b

Figure 11. The diffeomorphism dm .
For simplicity, we denote taij = ai,j , tbjk = bj,k , tckl = ck,l , tdm (aij ) = ai,j;m, tdm (bjk ) =
bj,k;m and tdm (ckl ) = ck,l;m. We remark that ai,j , bj,k and ck,l are described as shown in
Figure 12.

A NORMAL GENERATING SET FOR I(Ngb )

13

ai,j

bj,k

ck,l

Figure 12.
tδb , tρb and tσkb are explicitly described by products of Dehn twists as follows.
tδb = (tδ1 · · · tδb−1 )−g−b+3
(a1,2 · · · a1,g · b1,1 · · · b1,b−1 ) · · · (ag−1,g · bg−1,1 · · · bg−1,b−1 )(bg,1 · · · bg,b−1 )
(c1,2 · · · c1,b−1 ) · · · (cb−3,b−2 cb−3,b−1 )(cb−2,b−1 ),
tρb = (tδ1 · · · tδb−1 )−g−b+4
(a1,2 · · · a1,g−1 · b1,1 · · · b1,b−1 ) · · · (ag−2,g−1 · bg−2,1 · · · bg−2,b−1 )(bg−1,1 · · · bg−1,b−1 )
(c1,2 · · · c1,b−1 ) · · · (cb−3,b−2 cb−3,b−1 )(cb−2,b−1 ),
tσkb = (tδ1 · · · tδk−1 · tδk+1 · · · tδb−1 )−g−b+4
(a1,2 · · · a1,g · b1,1 · · · b1,k−1 · b1,k+1 · · · b1,b−1 ) · · ·
(ag−1,g · bg−1,1 · · · bg−1,k−1 · bg−1,k+1 · · · bg−1,b−1 )(bg,1 · · · bg,k−1 · bg,k+1 · · · bg,b−1 )
(c1,2 · · · c1,k−1 · c1,k+1 · · · c1,b−1 ) · · · (ck−2,k−1 · ck−2,k+1 · · · ck−2,b−1 )
(ck−1,k+1 · · · ck−1,b−1)(ck+1,k+2 · · · ck+1,b−1 ) · · · (cb−3,b−2 cb−3,b−1 )(cb−2,b−1 ).
In addition, Since tσ̄kb = t−1
dk (σkb ) , tσ̄kb is a product of ai,j;k , bi,j;k , ci,j;k and tδi .
Thus we obtain the claim.



Acknowledgement
The author would like to express his thanks to Genki Omori for his valuable suggestions
and useful comments.
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Department of General Education,
Ishikawa National College of Technology,
Tsubata, Ishikawa, 929-0392, Japan
E-mail address: kobayashi ryoma@ishikawa-nct.ac.jp

