arXiv:1509.05205v2 [math.GT] 10 Jul 2016

A FINITE GENERATING SET FOR THE LEVEL 2 TWIST
SUBGROUP OF THE MAPPING CLASS GROUP OF A CLOSED
NON-ORIENTABLE SURFACE
RYOMA KOBAYASHI AND GENKI OMORI

Abstract. We obtain a finite generating set for the level 2 twist subgroup of
the mapping class group of a closed non-orientable surface. The generating
set consists of crosscap pushing maps along non-separating two-sided simple
loops and squares of Dehn twists along non-separating two-sided simple closed
curves. We also prove that the level 2 twist subgroup is normally generated
in the mapping class group by a crosscap pushing map along a non-separating
two-sided simple loop for genus g ≥ 5 and g = 3. As an application, we
calculate the first homology group of the level 2 twist subgroup for genus
g ≥ 5 and g = 3.

1. Introduction
Let Ng,n be a compact connected non-orientable surface of genus g ≥ 1 with
n ≥ 0 boundary components. The surface Ng = Ng,0 is a connected sum of g
real projective planes. The mapping class group M(Ng,n ) of Ng,n is the group
of isotopy classes of self-diffeomorphisms on Ng,n fixing the boundary pointwise
and the twist subgroup T (Ng,n ) of M(Ng,n ) is the subgroup of M(Ng,n ) generated by all Dehn twists along two-sided simple closed curves. Lickorish [7] proved
that T (Ng ) is an index 2 subgroup of M(Ng ) and the non-trivial element of
M(Ng )/T (Ng ) ∼
= Z/2Z =: Z2 is represented by a “Y-homeomorphism”. We define
a Y-homeomorphism in Section 2. Chillingworth [1] gave an explicit finite generating set for T (Ng ) and showed that T (N2 ) ∼
= Z2 . The first homology group H1 (G)
of a group G is isomorphic to the abelianization Gab of G. The group H1 (T (Ng ))
is trivial for g ≥ 7, H1 (T (N3 )) ∼
= Z12 , H1 (T (N4 )) ∼
= Z2 ⊕ Z and H1 (T (Ng )) ∼
= Z2
for g = 5, 6. These results were shown by Korkmaz [5] for g ≥ 7 and by Stukow [10]
for the other cases.
Let Σg,n be a compact connected orientable surface of genus g ≥ 0 with n ≥ 0
boundary components. The mapping class group M(Σg,n ) of Σg,n is the group
of isotopy classes of orientation preserving self-diffeomorphisms on Σg,n fixing the
boundary pointwise. Let S be either Ng,n or Σg,n . For n = 0 or 1, we denote
by Γ2 (S) the subgroup of M(S) which consists of elements acting trivially on
H1 (S; Z2 ). Γ2 (S) is called the level 2 mapping class group of S. For a group G,
a normal subgroup H of G and a subset X of H, H is normally generated in G
by X if H is the normal closure of X in G. In particular, for X = {x1 , . . . , xn },
if H is the normal closure of X in G, we also say that H is normally generated
in G by x1 , . . . , xn . In the case of orientable surfaces, Humphries [3] proved that
Date: November 13, 2017.
2010 Mathematics Subject Classification. 57M05, 57M07, 57M20, 57M60.
1

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R. KOBAYASHI AND G. OMORI

Γ2 (Σg,n ) is normally generated in M(Σg,n ) by the square of the Dehn twist along
a non-separating simple closed curve for g ≥ 1 and n = 0 or 1. In the case of nonorientable surfaces, Szepietowski [11] proved that Γ2 (Ng ) is normally generated in
M(Ng ) by a Y-homeomorphism for g ≥ 2. Szepietowski [12] also gave an explicit
finite generating set for Γ2 (Ng ). This generating set is minimal for g = 3, 4. Hirose
and Sato [2] gave a minimal generating set for Γ2 (Ng ) when g ≥ 5 and showed that
g
g
∼ Z(3)+(2) .
H1 (Γ2 (Ng )) =
2
We denote by T2 (Ng ) the subgroup of T (Ng ) which consists of elements acting
trivially on H1 (Ng ; Z2 ) and we call T2 (Ng ) the level 2 twist subgroup of M(Ng ).
Recall that T (N2 ) ∼
= Z2 and Chillingworth [1] proved that T (N2 ) is generated by
the Dehn twist along a non-separating two-sided simple closed curve. T2 (N2 ) is
a trivial group because Dehn twists along non-separating two-sided simple closed
curves induce nontrivial actions on H1 (Ng ; Z2 ). Let Aut(H1 (Ng ; Z2 ), ·) be the group
of automorphisms on H1 (Ng ; Z2 ) preserving the intersection form · on H1 (Ng ; Z2 ).
Since the action of M(Ng ) on H1 (Ng ; Z2 ) preserves the intersection form ·, there
is the natural homomorphism from M(Ng ) to Aut(H1 (Ng ; Z2 ), ·). McCarthy and
Pinkall [8] showed that the restriction of the homomorphism to T (Ng ) is surjective.
Thus T2 (Ng ) is finitely generated.
In this paper, we give an explicit finite generating set for T2 (Ng ) (Theorem 3.1).
The generating set consists of “crosscap pushing maps” along non-separating twosided simple loops and squares of Dehn twists along non-separating two-sided simple
closed curves. We review the crosscap pushing map in Section 2. We can see the
generating set for T2 (Ng ) in Theorem 3.1 is minimal for g = 3 by Theorem 1.2. We
prove Theorem 3.1 in Section 3. In the last part of Subsection 3.2, we also give the
smaller finite generating set for T2 (Ng ) (Theorem 3.14). However, the generating
set consists of crosscap pushing maps along non-separating two-sided simple loops,
squares of Dehn twists along non-separating two-sided simple closed curves and
squares of Y-homeomorphisms.
By using the finite generating set for T2 (Ng ) in Theorem 3.1, we prove the
following theorem in Section 4.
Theorem 1.1. For g = 3 and g ≥ 5, T2 (Ng ) is normally generated in M(Ng ) by a
crosscap pushing map along a non-separating two-sided simple loop (See Figure 1).
T2 (N4 ) is normally generated in M(N4 ) by a crosscap pushing map along a
non-separating two-sided simple loop and the square of the Dehn twist along a nonseparating two-sided simple closed curve whose complement is a connected orientable
surface (See Figure 2).
The x-marks as in Figure 1 and Figure 2 mean Möbius bands attached to boundary components in this paper and we call the Möbius band the crosscap. The group
which is normally generated in M(Ng ) by the square of the Dehn twist along a
non-separating two-sided simple closed curve is a subgroup of T2 (Ng ) clearly. The
authors do not know whether T2 (Ng ) is generated by squares of Dehn twists along
non-separating two-sided simple closed curves or not.
As an application of Theorem 1.1, we calculate H1 (T2 (Ng )) for g ≥ 5 in Section 5
and we obtain the following theorem.

GENERATING LEVEL 2 TWIST SUBGROUP

3

Figure 1. A crosscap pushing map along a non-separating twosided simple loop is described by a product of Dehn twists along
non-separating two-sided simple closed curves as in the figure.

Figure 2. A non-separating two-sided simple closed curve on N4
whose complement is a connected orientable surface.
Theorem 1.2. For g = 3 and g ≥ 5, the first homology group of T2 (Ng ) is as
follows:
( 2
Z ⊕ Z2
if g = 3,
∼
H1 (T2 (Ng )) =
(g3)+(g2)−1
if g ≥ 5.
Z2
In this proof, we use the five term exact sequence for an extension of a group for
g ≥ 5. The authors do not know the first homology group of T2 (N4 ).
2. Preliminaries
2.1. Crosscap pushing map. Let S be a compact surface and let e : D′ ֒→ intS
be a smooth embedding of the unit disk D′ ⊂ C. Put D := e(D′ ). Let S ′ be the
surface obtained from S − intD by the identification of antipodal points of ∂D. We
call the manipulation that gives S ′ from S the blowup of S on D. Note that the
image M of the regular neighborhood of ∂D in S − intD by the blowup of S on
D is a crosscap, where a crosscap is a Möbius band in the interior of a surface.
Conversely, the blowdown of S ′ on M is the following manipulation that gives S
from S ′ . We paste a disk on the boundary obtained by cutting S along the center
line µ of M . The blowdown of S ′ on M is the inverse manipulation of the blowup
of S on D.
Let x0 be a point of Ng−1 and let e : D′ ֒→ Ng−1 be a smooth embedding of a
unit disk D′ ⊂ C to Ng−1 such that the interior of D := e(D′ ) contains x0 . Let
M(Ng−1 , x0 ) be the group of isotopy classes of self-diffeomorphisms on Ng−1 fixing
the point x0 , where isotopies also fix x0 . Then we have the blowup homomorphism
ϕ : M(Ng−1 , x0 ) → M(Ng )

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R. KOBAYASHI AND G. OMORI

that is defined as follows. For h ∈ M(Ng−1 , x0 ), we take a representative h′ of h
which satisfies either of the following conditions: (a) h′ |D is the identity map on D,
(b) h′ (x) = e(e−1 (x)) for x ∈ D. Such h′ is compatible with the blowup of Ng−1
on D, thus ϕ(h) ∈ M(Ng ) is induced and well defined (c.f. [11, Subsection 2.3]).
The point pushing map
j : π1 (Ng−1 , x0 ) → M(Ng−1 , x0 )
is a homomorphism that is defined as follows. For γ ∈ π1 (Ng−1 , x0 ), j(γ) ∈
M(Ng−1 , x0 ) is described as the result of pushing the point x0 once along γ. Note
that for x, y ∈ π1 (Ng−1 ), yx means yx(t) = x(2t) for 0 ≤ t ≤ 21 and yx(t) = y(2t−1)
for 12 ≤ t ≤ 1, and for elements [f ], [g] of the mapping class group, [f ][g] means
[f ◦ g].
We define the crosscap pushing map as the composition of homomorphisms:
ψ := ϕ ◦ j : π1 (Ng−1 , x0 ) → M(Ng ).
For γ ∈ π1 (Ng−1 , x0 ), we also call ψ(γ) the crosscap pushing map along γ. Remark
that for γ, γ ′ ∈ π1 (Ng−1 , x0 ), ψ(γ)ψ(γ ′ ) = ψ(γγ ′ ). The next two lemmas follow
from the description of the point pushing map (See [6, Lemma 2.2, Lemma 2.3]).
Lemma 2.1. For a two-sided simple loop γ on Ng−1 based at x0 , suppose that γ1 ,
γ2 are two-sided simple closed curves on Ng−1 such that γ1 ⊔ γ2 is the boundary
of the regular neighborhood N of γ in Ng−1 whose interior contains D. Then for
some orientation of N , we have
t−1
,
ψ(γ) = ϕ(tγ1 t−1
f
γ 2 ) = tγ
1 γ
f
2
where γe1 , γe2 are images of γ1 , γ2 to Ng by blowups respectively (See Figure 3).

Let µ be a one-sided simple closed curve and let α be a two-sided simple closed
curve on Ng such that µ and α intersect transversely at one point. For these simple
closed curves µ and α, we denote by Yµ,α a self-diffeomorphism on Ng which is
described as the result of pushing the regular neighborhood of µ once along α.
We call Yµ,α a Y-homeomorphism (or crosscap slide). By Lemma 3.6 in [11], Yhomeomorphisms are in Γ2 (Ng ).
Lemma 2.2. Suppose that γ is a one-sided simple loop on Ng−1 based at x0 such
that γ and ∂D intersect at antipodal points of ∂D. Then we have
ψ(γ) = Yµ,eγ ,
where γ
e is a image of γ to Ng by a blowup and µ is a center line of the crosscap
obtained from the regular neighborhood of ∂D in Ng−1 by the blowup of Ng−1 on D
(See Figure 4).
Remark that the image of a crosscap pushing map is contained in Γ2 (Ng ). By
Lemma 2.1, if γ is a two-sided simple loop on Ng , then ψ(γ) is an element of T2 (Ng ).
We remark that Y-homeomorphisms are not in T (Ng ) (See [7]).
2.2. Notation of the surface Ng . Let ei : Di′ ֒→ Σ0 for i = 1, 2, . . . , g be smooth
embeddings of unit disks Di′ ⊂ C to a 2-sphere Σ0 such that Di := ei (Di′ ) and
Dj are disjoint for distinct 1 ≤ i, j ≤ g, and let xi ∈ Σ0 for i = 1, 2, . . . , g be g
points of Σ0 such that xi is contained in the interior of Di as the left-hand side
of Figure 5. Then Ng is diffeomorphic to the surface obtained from Σ0 by the
blowups on D1 , . . . , Dg . We describe the identification of ∂Di by the x-mark as

GENERATING LEVEL 2 TWIST SUBGROUP

5

Figure 3. A crosscap pushing map along two-sided simple loop γ.

Figure 4. A crosscap pushing map along one-sided simple loop γ
(Y-homeomorphism Yµ,eγ ).
the right-hand side of Figure 5. We call the crosscap which is obtained from the
regular neighborhood of ∂Di in Σ0 by the blowup of Σ0 on Di the i-th crosscap.
(k)
We denote by Ng−1 the surface obtained from Σ0 by the blowups on Di for every
(k)

i 6= k. Ng−1 is diffeomorphic to Ng−1 . Let xk;i be a simple loop on Ng based at
(k)

(k)

xk for i 6= k as Figure 6. Then the fundamental group π1 (Ng−1 ) = π1 (Ng−1 , xk ) of
(k)

Ng−1 has the following presentation.
(k)

π1 (Ng−1 ) = xk;1 , . . . , xk;k−1 , xk;k+1 , . . . , xk;g | x2k;1 . . . x2k;k−1 x2k;k+1 . . . x2k;g = 1 .

Figure 5. The embedded disks D1 , D2 , . . . , Dg on Σ0 and the
surface Ng .
(k)

2.3. Notations of mapping classes. Let ψk : π1 (Ng−1 ) → M(Ng ) be the cross(k)

(k)

cap pushing map obtained from the blowup of Ng−1 on Dk and let π1 (Ng−1 )+ be

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R. KOBAYASHI AND G. OMORI

Figure 6. The simple loop xk;i for 1 ≤ i ≤ k − 1 and xk;j for
(k)
k + 1 ≤ j ≤ g on Ng−1 based at xk .
(k)

(k)

the subgroup of π1 (Ng−1 ) generated by two-sided simple loops on Ng−1 based at
(k)

xk . By Lemma 2.1, we have ψk (π1 (Ng−1 )+ ) ⊂ T2 (Ng ). We define non-separating
(k)

two-sided simple loops αk;i,j and βk;i,j on Ng−1 based at xk as in Figure 7 for distinct 1 ≤ i < j ≤ g and 1 ≤ k ≤ g. We also define αk;j,i := αk;i,j and βk;j,i := βk;i,j
for distinct 1 ≤ i < j ≤ g and 1 ≤ k ≤ g. We have the following equations:
αk;i,j = xk;i xk;j

for i < j < k or j < k < i or k < i < j,

βk;i,j = xk;j xk;i

for i < j < k or j < k < i or k < i < j.

Denote the crosscap pushing maps ak;i,j := ψk (αk;i,j ) and bk;i,j := ψk (βk;i,j ). Remark that ak;i,j and bk;i,j are contained in the image of ψk |π1 (N (k) )+ . Let η be the
g−1

self-diffeomorphism on Ng which is the rotation of Ng such that η sends the i-th
crosscap to the (i + 1)-st crosscap for 1 ≤ i ≤ g − 1 and the g-th crosscap to the
1-st crosscap as Figure 8. Then we have ak;i,j = η k−1 a1;i−k+1,j−k+1 η −(k−1) and
bk;i,j = η k−1 b1;i−k+1,j−k+1 η −(k−1) for each distinct 1 ≤ i, j, k ≤ g.

(k)

Figure 7. Two-sided simple loops αk;i,j and βk;i,j on Ng−1 based
at xk .
For distinct i1 , i2 , . . . , in ∈ {1, 2, . . . , g}, we define a simple closed curve αi1 ,i2 ,...,in
on Ng as in Figure 9. The arrow on the side of the simple closed curve αi1 ,i2 ,...,in

GENERATING LEVEL 2 TWIST SUBGROUP

7

Figure 8. The self-diffeomorphism η on Ng .
in Figure 9 indicates the direction of the Dehn twist tαi1 ,i2 ,...,in along αi1 ,i2 ,...,in if
n is even. We set the notations of Dehn twists and Y-homeomorphisms as follows:
for 1 ≤ i < j ≤ g,

Ti,j

:=

tαi,j

Ti,j,k,l

:=

tαi,j,k,l

Yi,j

:=

Yαi ,αi,j = ψi (xi;j )

for g ≥ 4 and 1 ≤ i < j < k < l ≤ g,
for distinct 1 ≤ i, j ≤ g.

2
2
Note that Ti,j
and Ti,j,k,l
are elements of T2 (Ng ), Yi,j is an element of Γ2 (Ng ) but
2
Yi,j is not an element of T2 (Ng ). We remark that ak;i,j = b−1
k;i,j = Ti,j for any
distinct i, j, k ∈ {1, 2, 3} when g = 3.

Figure 9. The simple closed curve αi1 ,i2 ,...,in on Ng .

3. Finite generating set for T2 (Ng )
In this section, we prove the main theorem in this paper. The main theorem is
as follows:
Theorem 3.1. For g ≥ 3, T2 (Ng ) is generated by the following elements:
(i) ak;i,i+1 , bk;i,i+1 , ak;k−1,k+1 , bk;k−1,k+1 for 1 ≤ k ≤ g, 1 ≤ i ≤ g and
i 6= k − 1, k,
(ii) a1;2,4 , bk;1,4 , al;1,3 for k = 2, 3 and 4 ≤ l ≤ g when g is odd,
2
(iii) T1,j,k,l
for 2 ≤ j < k < l ≤ g when g ≥ 4,
where the indices are considered modulo g.
We remark that the number of generators in Theorem 3.1 is 16 (g 3 + 6g 2 + 5g − 6)
for g ≥ 4 odd, 16 (g 3 + 6g 2 − g − 6) for g ≥ 4 even and 3 for g = 3.

8

R. KOBAYASHI AND G. OMORI
(k)

3.1. Finite generating set for π1 (Ng−1 )+ . First, we have the following lemma:
(k)

(k)

Lemma 3.2. For g ≥ 2, π1 (Ng−1 )+ is an index 2 subgroup of π1 (Ng−1 ).
(k)

Proof. Note that π1 (Ng−1 ) is generated by xk;1 , . . . , xk;k−1 , xk;k+1 , . . . , xk;g . If g =
(k)

2, π1 (Ng−1 ) is isomorphic to Z2 which is generated by a one-sided simple loop.
(k)

Hence π1 (Ng−1 )+ is trivial and we obtain this lemma when g = 2.
We assume that g ≥ 3. For i 6= k, we have
xk;i = x−1
k;k−1 · xk;k−1 xk;i .
(k)

Since xk;k−1 xk;i = βk;i,k−1 ∈ π1 (Ng−1 )+ , the equivalence classes of xk;i and x−1
k;k−1
(k)

(k)

in π1 (Ng−1 )/π1 (Ng−1 )+ are the same. We also have
2
xk;k−1 = x−1
k;k−1 · xk;k−1 .
(k)

Since x2k;k−1 ∈ π1 (Ng−1 )+ , the equivalence classes of xk;k−1 and x−1
k;k−1 in
(k)

(k)

(k)

(k)

π1 (Ng−1 )/π1 (Ng−1 )+ are the same. Thus π1 (Ng−1 )/π1 (Ng−1 )+ is generated by
the equivalence class [xk;k−1 ] whose order is 2 and we have completed the proof of
Lemma 3.2.

(k)

Ng−1 is diffeomorphic to the surface on the left-hand side (resp. right-hand
side) of Figure 10 when g − 1 = 2h + 1 (resp. g − 1 = 2h + 2). We take a
diffeomorphism which sends xk;i for i 6= k and xk as in Figure 6 to xk;i for i 6= k
(k)
and xk as in Figure 10 and identify Ng−1 with the surface in Figure 10 by the
]
(k)
(k)
diffeomorphism. Denote by pk : Ng−1 ։ Ng−1 the orientation double covering
]
(k)
(k)
of Ng−1 as in Figure 11. Then Hk := (pk )∗ (π1 (Ng−1 )) is an index 2 subgroup
]
(k)
(k)
of π1 (Ng−1 ). Note that when g − 1 = 2h + 1, π1 (Ng−1 ) is generated by yk;i for
]
(k)
1 ≤ i ≤ 4h, and when g−1 = 2h+2, π1 (Ng−1 ) is generated by yk;i for 1 ≤ i ≤ 4h+2,
]
(k)
where yk;i is two-sided simple loops on Ng−1 based at the lift x
fk of xk as in Figure 11.
We have the following Lemma.
Lemma 3.3. For g − 1 ≥ 1 and 1 ≤ k ≤ g,

(k)

Hk = π1 (Ng−1 )+ .
(k)

(k)

Proof. Note that π1 (Ng−1 )+ is an index 2 subgroup of π1 (Ng−1 ) by Lemma 3.2. It
(k)

is sufficient for proof of Lemma 3.3 to prove Hk ⊂ π1 (Ng−1 )+ because the index of
Hk in

(k)
π1 (Ng−1 )

is
(k)

(k)

(k)

(k)

2 = [π1 (Ng−1 ) : Hk ] = [π1 (Ng−1 ) : π1 (Ng−1 )+ ][π1 (Ng−1 )+ : Hk ]
(k)

= 2 · [π1 (Ng−1 )+ : Hk ]
(k)

if Hk ⊂ π1 (Ng−1 )+ .
(k)

We define subsets of π1 (Ng−1 )+ as follows:
A :=

{xk;j+1 xk;j , xk;k+1 xk;k−1 | 1 ≤ j ≤ g − 1, j 6= k − 1, k},

B

{xk;j xk;j+1 , xk;k−1 xk;k+1 | 1 ≤ j ≤ g − 1, j 6= k − 1, k},

:=

GENERATING LEVEL 2 TWIST SUBGROUP

9

(k)

Figure 10. Ng−1 is diffeomorphic to the surface on the left-hand
side (resp. right-hand side) of the figure when g − 1 = 2h + 1 (resp.
g−1 = 2h+2). We regard the above surface on the left-hand side as
the surface identified antipodal points of the boundary component,
and the above surface on the right-hand side as the surface attached
their boundary components along the orientation of the boundary.
C

:=



{x2k;1 }
{x2k;2 }

if k 6= 1,
if k = 1.

]
(k)
π1 (Ng−1 ) is generated by yk;i . For i ≤ 2h when g − 1 = 2h + 1 (resp. i ≤ 2h + 1
when g − 1 = 2h + 2), we can check that

x
x
if 2 ≤ i ≤ 2h and g − 1 = 2h + 1,


 k;ρ(i+1) k;ρ(i)
xk;g xk;g−1
if i = 1 and g − 1 = 2h + 1,
(pk )∗ (yk;i ) =
x
x
if 2 ≤ i ≤ 2h + 1 and g − 1 = 2h + 2,


 k;ρ(i+1) k;ρ(i)
xk;g xk;g−1
if i = 1 and g − 1 = 2h + 2,

and (pk )∗ (yk;i ) is an element of A, where ρ is the order reversing bijection from
{1, 2, . . . , 2h} (resp. {1, 2, . . . , 2h+1}) to {1, 2, . . . , g−1}−{k}. Since if g−1 = 2h+1
and g − 1 = 2h′ + 2, we have
 2
xk;1 if k 6= 1,
(pk )∗ (yk;2h+1 ) =
x2k;2 if k = 1,
 2
xk;1 if k 6= 1,
(pk )∗ (yk;2h′ +2 ) =
x2k;2 if k = 1,
(pk )∗ (yk;2h+1 ) and (pk )∗ (yk;2h′ +2 ) are elements of C respectively (See Figure 12).
Finally, for i ≥ 2h + 2 when g − 1 = 2h + 1 (resp. i ≥ 2h + 3 when g = 2h + 2), we
can also check that

x ′ x ′
if 2h + 2 ≤ i ≤ 4h − 1 and g − 1 = 2h + 1,


 k;ρ (i) k;ρ (i+1)
xk;g−1 xk;g
if i = 4h and g − 1 = 2h + 1,
(pk )∗ (yk;i ) =
′ (i) xk;ρ′ (i+1)
x
if 2h + 3 ≤ i ≤ 4h + 1 and g − 1 = 2h + 2,

k;ρ


xk;g−1 xk;g
if i = 4h + 2 and g − 1 = 2h + 2,

and (pk )∗ (yk;i ) = xk;ρ′ (i) xk;ρ′ (i+1) is an element of B, where ρ′ is the order preserving bijection from {2h + 2, 2h + 3, . . . , 4h} (resp. {2h + 3, 2h + 4, . . . , 4h + 2}) to
{1, 2, . . . , g − 1} − {k} (See Figure 13). We obtain this lemma.

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R. KOBAYASHI AND G. OMORI



]
(k)
Figure 11. The total space Ng−1 of the orientation double cov]
(k)
(k)
ering pk of Ng−1 and two-sided simple loops yk;i on Ng−1 based at
x
fk .

Figure 12. The representative of x2k;1 when k 6= 1 or x2k;2 when
k = 1.
By the proof of Lemma 3.3, we have the following proposition.
(k)

Proposition 3.4. For g ≥ 2, π1 (Ng−1 )+ is generated by the following elements:

GENERATING LEVEL 2 TWIST SUBGROUP

11

Figure 13. The representative of xk;j xk;j+1 and xk;k−1 xk;k+1 for
j 6= k − 1, k.
(1) xk;i+1 xk;i , xk;i xk;i+1 , xk;k+1 xk;k−1 , xk;k−1 xk;k+1 for 1 ≤ i ≤ g − 1 and
i 6= k − 1, k,
(2) x2k;2 when k = 1,
(3) x2k;1 when 2 ≤ k ≤ g.
We remark that xk;i+1 xk;i = βk;i,i+1 , xk;i xk;i+1 = αk;i,i+1 , xk;k+1 xk;k−1 =
2
2
αk;k−1,k+1 , xk;k−1 xk;k+1 = βk;k−1,k+1 and Yi,j
= Yj,i
. Let G be the subgroup of
(k)

T2 (Ng ) generated by ∪gk=1 ψk (π1 (Ng−1 )+ ). The next corollary follows from Proposition 3.4 immediately.
Corollary 3.5. For g ≥ 2, G is generated by the following elements:
(i) ak;i,i+1 , bk;i,i+1 , ak;k−1,k+1 , bk;k−1,k+1 for 1 ≤ k ≤ g, 1 ≤ i ≤ g − 1 and
i 6= k − 1, k,
2
(ii) Y1,j
when 2 ≤ j ≤ g,
where the indices are considered modulo g.
The simple loop x2k;1 and x2k;2 are separating loops. By the next proposition,
(k)

π1 (Ng−1 )+ is generated by finitely many two-sided non-separating simple loops.
(k)

Proposition 3.6. For g ≥ 2, π1 (Ng−1 )+ is generated by the following elements:
(1) xk;i+1 xk;i , xk;i xk;i+1 , xk;k+1 xk;k−1 , xk;k−1 xk;k+1 for 1 ≤ i ≤ g and i 6=
k − 1, k,
(2) xk;2 xk;4 when k = 1 and g − 1 is even,
(3) xk;1 xk;4 when k = 2, 3 and g − 1 is even,
(4) xk;1 xk;3 when 4 ≤ k ≤ g and g − 1 is even,
where the indices are considered modulo g.
Proof. When g − 1 is odd, since we have
−1
−1
−1
x21;2 = x1;2 x1;3 · x−1
1;3 x1;4 · x1;4 x1;5 · · · · · x1;g−1 x1;g · x1;g x1;2

12

R. KOBAYASHI AND G. OMORI

and
−1
−1
−1
x2k;1 = xk;1 xk;2 · x−1
k;2 xk;3 · xk;3 xk;4 · · · · · xk;g−1 xk;g · xk;g xk;1

for 2 ≤ k ≤ g, this proposition is clear.
When g − 1 is even, we use the relation
x2k;1 . . . x2k;k−1 x2k;k+1 . . . x2k;g = 1.
For k = 1, we have
x21;2 = x1;2 x1;4 · x1;4 x1;5 · · · · · x1;g x1;2 · x1;2 x1;3 · x1;3 x1;2 .
By a similar argument, we also have following equations:

x2 = x2;1 x2;4 · x2;4 x2;5 · · · · · x2;g x2;1 · x2;1 x2;3 · x2;3 x2;1 if k = 2,


 22;1

x3;1 = x3;1 x3;4 · x3;4 x3;5 · · · · · x3;g x3;1 · x3;1 x3;2 · x3;2 x3;1 if k = 3,
x2k;1 =
x2k;1 = xk;1 xk;3 · xk;3 xk;4 · · · · · xk;k−1 xk;k+1 · · · ·




·xk;g xk;1 · xk;1 xk;2 · xk;2 xk;1
if 4 ≤ k ≤ g.

We obtain this proposition.



We remark that xk;1 xk;g = βk;1,g , xk;g xk;1 = αk;1,g , x1;2 x1;4 = α1;2,4 , xk;1 xk;4 =
βk;1,4 for k = 2, 3 and xk;1 xk;3 = αk;1,3 for 4 ≤ k ≤ g. By the above remarks, we
have the following corollary.
Corollary 3.7. For g ≥ 2, G is generated by the following elements:
(i) ak;i,i+1 , bk;i,i+1 , ak;k−1,k+1 , bk;k−1,k+1 for 1 ≤ k ≤ g, 1 ≤ i ≤ g and
i 6= k − 1, k,
(ii) a1;2,4 , bk;1,4 , al;1,3 for k = 2, 3 and 4 ≤ l ≤ g when g is odd,
where the indices are considered modulo g.
3.2. Proof of Main-Theorem. First, we obtain a finite generating set for T2 (Ng )
by the Reidemeister-Schreier method. We use the following minimal generating set
for Γ2 (Ng ) given by Hirose and Sato [2] when g ≥ 5 and Szepietowski [12] when
g = 3, 4 to apply the Reidemeister-Schreier method. See for instance [4] to recall
the Reidemeister-Schreier method.
Theorem 3.8. [2, 12] For g ≥ 3, Γ2 (Ng ) is generated by the following elements:
(1) Yi,j for 1 ≤ i ≤ g − 1, 1 ≤ j ≤ g and i 6= j,
2
(2) T1,j,k,l
for 2 ≤ j < k < l ≤ g when g ≥ 4.
Proposition 3.9. For g ≥ 3, T2 (Ng ) is generated by the following elements:
2
(1) Yi,j Y1,2 , Yi,j
for 1 ≤ i ≤ g − 1, 1 ≤ j ≤ g and i 6= j,
−1 2
2
for 2 ≤ j < k < l ≤ g when g ≥ 4.
(2) Y1,2 T1,j,k,l Y1,2 , T1,j,k,l
Proof. Note that T2 (Ng ) is the intersection of Γ2 (Ng ) and T (Ng ). Hence we have
the isomorphisms
Γ2 (Ng )/(Γ2 (Ng ) ∩ T (Ng )) ∼
= Z2 [Y1,2 ].
= (Γ2 (Ng )T (Ng ))/T (Ng ) ∼
We remark that Γ2 (Ng )T (Ng ) = M(Ng ) and the last isomorphism is given by
Lickorish [7]. Thus T2 (Ng ) is an index 2 subgroup of Γ2 (Ng ).
Set U := {Y1,2 , 1} and X as the generating set for Γ2 (Ng ) in Theorem 3.8, where
1 means the identity element. Then U is a Schreier transversal for T2 (Ng ) in Γ2 (Ng ).

GENERATING LEVEL 2 TWIST SUBGROUP

13

For x ∈ Γ2 (Ng ), define x as the element of U such that [x] = [x] in Γ2 (Ng )/T2 (Ng ).
By the Reidemeister-Schreier method, for g ≥ 4, T2 (Ng ) is generated by
B ={wu−1 wu | w ∈ X ± , u ∈ U, wu 6∈ U }
±1
Y1,2
={Yi,j

−1

±1
±1
Yi,j
Y1,2 , Yi,j

±2
Y1,2
∪ {T1,j,k,l

−1

−1

±1
Yi,j
| 1 ≤ i ≤ g − 1, 1 ≤ j ≤ g, i 6= j}

±2
±2
T1,j,k,l
Y1,2 , T1,j,k,l

−1

±2
T1,j,k,l
| 2 ≤ j < k < l ≤ g}

±1
−1 ±1
={Yi,j
Y1,2 , Y1,2
Yi,j | 1 ≤ i ≤ g − 1, 1 ≤ j ≤ g, i 6= j}
−1 ±2
±2
∪ {Y1,2
T1,j,k,l Y1,2 , T1,j,k,l
| 2 ≤ j < k < l ≤ g},

where X ± := X ∪ {x−1 | x ∈ X} and note that equivalence classes of Y−1 ±1
∓1
Y1,2 )−1
homeomorphisms in Γ2 (Ng )/T2 (Ng ) is nontrivial. Since Y1,2
Yi,j = (Yi,j
−1
−2
and Yi,j Y1,2 = Yi,j · Yi,j Y1,2 , we have the following generating set for T2 (Ng ):
2
B ′ ={Yi,j Y1,2 , Yi,j
| 1 ≤ i ≤ g − 1, 1 ≤ j ≤ g, i 6= j}
−1 2
2
∪ {Y1,2
T1,j,k,l Y1,2 , T1,j,k,l
| 2 ≤ j < k < l ≤ g}.

By a similar discussion, T2 (N3 ) is generated by
2
B ′ ={Yi,j Y1,2 , Yi,j
| 1 ≤ i ≤ 2, 1 ≤ j ≤ 3, i 6= j}.

We obtain this proposition.



Let G be the group generated by the elements of type (i), (ii) and (iii) in Theorem 3.1. Then G is a subgroup of T2 (Ng ) clearly and it is sufficient for the proof
of Theorem 3.1 to prove B ′ ⊂ G, where B ′ is the generating set for T2 (Ng ) in the
(k)
proof of Proposition 3.9. By Corollary 3.7, we have ψk (π1 (Ng−1 )+ ) ⊂ G for any
(i)

2
1 ≤ k ≤ g. Thus Yi,j
= ψi (x2i;j ) ∈ ψi (π1 (Ng−1 )+ ) ⊂ G. We complete the proof of
−1 2
Theorem 3.1 if Yi,j Y1,2 and Y1,2
T1,j,k,l Y1,2 are in G.
−1 2
Lemma 3.10. For g ≥ 4, Y1,2
T1,j,k,l Y1,2 ∈ G.
−1 2
−1 2
Proof. Since Y1,2
T1,j,k,l Y1,2 is a Dehn twist along
T1,j,k,l Y1,2 = tY −1 (α1,j,k,l ) , Y1,2
1,2
the two-sided simple closed curve as in Figure 14. Then we have a1;k,l (α1,2,k,l ) =
−1
−2
−1
Y1,2
(α1,2,k,l ) and Y1,2
a1;2,j a1;k,l (α1,j,k,l ) = Y1,2
(α1,j,k,l ) for 3 ≤ j ≤ g
and the local orientation of the regular neighborhood of a1;k,l (α1,2,k,l ) (resp.
−2
−1
−1
Y1,2
a1;2,j a1;k,l (α1,j,k,l )) and Y1,2
(α1,2,k,l ) (resp. Y1,2
(α1,j,k,l )) are different. Therefore we have
−1 2
−2
Y1,2
T1,2,k,l Y1,2 =a1;k,l T1,2,k,l
a−1
1;k,l ,
−1 2
−2
−2
−1
2
Y1,2
T1,j,k,l Y1,2 =Y1,2
a1;2,j a1;k,l T1,j,k,l
a−1
1;k,l a1;2,j Y1,2 for 3 ≤ j ≤ g.
(1)

By Corollary 3.7, a1;k,l , a1;2,j ∈ ψ1 (π1 (Ng−1 )+ ) ⊂ G. We obtain this lemma.



Szepietowski [11, Lemma 3.1] showed that for any non-separating two-sided simple closed curve γ, t2γ is a product of two Y-homeomorphisms. In particular, we
have the following lemma.
Lemma 3.11 ([11]). For distinct 1 ≤ i, j ≤ g,
 2
Ti,j for i < j,
−1
−1
Yj,i
Yi,j = Yj,i Yi,j
=
−2
Ti,j
for j < i.

14

R. KOBAYASHI AND G. OMORI

Figure 14. The upper side of the figure is the simple closed curve
−1
Y1,2
(α1,2,k,l ) on Ng and the lower side of the figure is the simple
−1
closed curve Y1,2
(α1,j,k,l ) on Ng for 3 ≤ j ≤ g.
2
Lemma 3.12. For distinct 1 ≤ i, j ≤ g, Ti,j
∈ G.

Proof. We discuss by a similar argument in proof of Lemma 3.5 in [12]. Let γi be
(i)
the two-sided simple loop on Ng−1 for i = 3, . . . , g as in Figure 15. Then we have
(i)

2
T1,2
= ψg (γg ) · · · ψ4 (γ4 )ψ3 (γ3 ) (see Figure 15). Since γi ∈ π1 (Ng−1 )+ , each ψi (γi )
2
is an element of G by Corollary 3.7. Hence we have T1,2
∈ G.
We denote by σi,j the self-diffeomorphism on Ng which is obtained by the transposition of the i-th crosscap and the j-th crosscap as in Figure 16. σi,j is called the
crosscap transposition (c.f. [9]). For 1 ≤ i < j ≤ g, put fi,j ∈ M(Ng ) as follows:

f1,2 := 1,
f1,j := σj−1,j · · · σ3,4 σ2,3 for 3 ≤ j ≤ g,
fi,j := σi−1.i · · · σ2,3 σ1,2 f1,j for 2 ≤ i < j ≤ g.
−1
−1
−1
−1
2
2
Then Ti,j
= fi,j T1,2
fi,j
= fi,j ψg (γg )fi,j
· · · fi,j ψ4 (γ4 )fi,j
·fi,j ψ3 (γ3 )fi,j
. Since the
−1
action of σi,j on Ng preserves the set of i-th crosscaps for 1 ≤ i ≤ g, fi,j ψk (γk )fi,j
is
(k′ )

−1
∈
an element of ψk′ (π1 (Ng−1 )) for some k ′ . By Corollary 3.7, we have fi,j ψk (γk )fi,j
G and we obtain this lemma.


Finally, by the following proposition, we complete the proof of Theorem 3.1.
Proposition 3.13. For distinct 1 ≤ i, j, k, l ≤ g, Yk,l Yi,j ∈ G.
Proof. Yk,l Yi,j is the following product of elements of G.
(a) case (k, l) = (i, j):
2
Yk,l Yi,j = Yi,j
.
By Corollary 3.7, the right-hand side is an element of G.
(b) case (k, l) = (j, i):
−1
2
Yj,i Yi,j = Yj,i
· Yj,i
Yi,j

Lem. 3.11

=

±2
(a) · Ti,j
.

By Lemma 3.12, the right-hand side is an element of G.
(c) case k = i and l 6= j:
Yi,l Yi,j = ψi (xi;l )ψi (xi;j ) = ψi (αi;j,l ) = ai;j,l

Cor. 3.7

∈

G.

GENERATING LEVEL 2 TWIST SUBGROUP

15

2
Figure 15. T1,2
is a product of crosscap pushing maps along γ3 , γ4 , . . . , γg .

Figure 16. The crosscap transposition σi,j .
(d) case k 6= i and l = j:
−1 −1
Yk,j Yi,j = Yk,j Yk,i · Yk,i
Yi,k · Yi,k Yi,j = (c) · (b) · (c) ∈ G.

(e) case k = j and l 6= i:
−1
Yj,l Yi,j = Yj,l Yj,i · Yj,i
Yi,j

Lem. 3.11

±2
(c) · Ti,j

Lem. 3.12

Lem. 3.11

±2
Ti,k
· (c)

Lem. 3.12

=

∈

G.

(f) case k 6= j and l = i:
−1
Yk,i Yi,j = Yk,i Yi,k
· Yi,k Yi,j

=

∈

G.

(g) case {k, l} ∩ {i, j} is empty:
−1 −1
Yk,l Yi,j = Yk,l Yk,j · Yk,j
Yk,j · Yk,j Yi,j = (c) · (a) · (d) ∈ G.

We have completed this proposition.



By a similar discussion in Subsection 3.2 and Corollary 3.5, we obtain the following theorem.
Theorem 3.14. For g ≥ 3, T2 (Ng ) is generated by following elements:
(i) ak;i,i+1 , bk;i,i+1 , ak;k−1,k+1 , bk;k−1,k+1 for 1 ≤ k ≤ g, 1 ≤ i ≤ g − 1 and
i 6= k − 1, k,

16

R. KOBAYASHI AND G. OMORI
2
(ii) Y1,j
for 2 ≤ j ≤ g,
2
(iii) T1,j,k,l for 2 ≤ j < k < l ≤ g when g ≥ 4,

where the indices are considered modulo g.
Since the number of generators in Theorem 3.14 is 61 (g 3 + 6g 2 − 7g − 12) for
g ≥ 4 and 3 for g = 3, the number of generators in Theorem 3.14 is smaller than
the number of generators in Theorem 3.1. On the other hand, by Theorem

 1.2,
the dimension of the first homology group H1 (T2 (Ng )) of T2 (Ng ) is g3 + g2 − 1 =
1 3
2
6 (g − g − 6) for g ≥ 4. The difference of them is g − g − 1. The authors do not
know the minimal number of generators for T2 (Ng ) when g ≥ 4.
4. Normal generating set for T2 (Ng )
The next lemma is a generalization of the argument in the proof of Lemma 3.5
in [12].
Lemma 4.1. Let γ be a non-separating two-sided simple closed curve on Ng such
that Ng − γ is a non-orientable surface. Then t2γ is a product of crosscap pushing
maps along two-sided non-separating simple loops such that their crosscap pushing
maps are conjugate to a1;2,3 in M(Ng ).
Proof of Theorem 1.1. By Theorem 3.1, T2 (Ng ) is generated by (I) crosscap push2
ing maps along non-separating two-sided simple loops and (II) T1,j,k,l
for 2 ≤ j <
2
2
2
2
k < l ≤ g. When g = 3, T2 (Ng ) is generated by T1,2 , T1,3 , T2,3 . Recall Ti,j
= a−1
k;i,j
when g = 3. Since Ng − αi,j is non-orientable for g ≥ 3, ak;i,j is conjugate to ak′ ;i′ ,j ′
in M(Ng ). Hence Theorem 1.1 is clear when g = 3.
(k)
Assume g ≥ 4. For a non-separating two-sided simple loop c on Ng−1 based at
xk , by Lemma 2.1, there exist non-separating two-sided simple closed curves c1 and
c2 such that ψ(c) = tc1 t−1
c2 , where c1 and c2 are images of boundary components of
(k)
regular neighborhood of c in Ng−1 to Ng by a blowup. Then the surface obtained
by cutting Ng along c1 and c2 is diffeomorphic to a disjoint sum of Ng−3,2 and
N1,2 . Thus mapping classes of type (I) is conjugate to a1;2,3 in M(Ng ). We obtain
Theorem 1.1 for g = 4.
Assume g ≥ 5. Simple closed curves αi,j,k,l satisfy the condition of Lemma 4.1.
2
Therefore T1,j,k,l
is a product of crosscap pushing maps along non-separating twosided simple loops and such crosscap pushing maps are conjugate to a1;2,3 in
M(Ng ). We have completed the proof of Theorem 1.1.

5. First homology group of T2 (Ng )
By the argument in the proof of Proposition 3.9, for g ≥ 2, we have the following
exact sequence:
(5.1)

1 −→ T2 (Ng ) −→ Γ2 (Ng ) −→ Z2 −→ 0,

where Z2 is generated by the equivalence class of a Y-homeomorphism.
The level 2 principal congruence subgroup Γ2 (n) of GL(n, Z) is the kernel of the
natural surjection GL(n, Z) ։ GL(n, Z2 ). Szepietowski [12, Corollary 4.2] showed
that there exists an isomorphism θ : Γ2 (N3 ) → Γ2 (2) which is induced by the action
of Γ2 (N3 ) on the free part of H1 (N3 ; Z). Since the determinant of the action of a

GENERATING LEVEL 2 TWIST SUBGROUP

17

Dehn twist on the free part of H1 (N3 ; Z) is 1, we have the following commutative
diagram of exact sequences:
/ Γ2 (N3 )

/ T2 (N3 )

1

(5.2)

θ|T2 (N3 )

/1


/ Z2

/ 1,

θ


/ SL(2, Z)[2]

1

/ Z2


/ Γ2 (2)

det

where SL(n, Z)[2] := Γ2 (n) ∩ SL(n, Z) is the level 2 principal congruence subgroup
of the integral special linear group SL(n, Z). By the commutative diagram (5.2),
T2 (N3 ) is isomorphic to SL(2, Z)[2].
Proof of Theorem 1.2. For g = 3, the first homology group H1 (T2 (N3 )) is isomorphic to H1 (SL(2, Z)[2]) by the commutative diagram (5.2). The restriction of the
natural surjection from SL(2, Z) to the projective special linear group P SL(2, Z)
to SL(n, Z)[2] gives the following commutative diagram of exact sequences:
(5.3)

1

/ Z2 [−E]

/ SL(2, Z)

/ P SL(2, Z)

/1


/ SL(2, Z)[2]


/ P SL(2, Z)[2]

/ 1,

id

1


/ Z2 [−E]

where E is the identity matrix and P SL(n, Z)[2] := SL(n, Z)/{±E} is the level 2
principal congruence subgroup of P SL(2, Z). Since P SL(2, Z)[2] is isomorphic to
the free group F2 of rank 2 and −E commutes with all matrices, the exact sequence
in the lower row of Diagram (5.3) is split and SL(2, Z)[2] is isomorphic to F2 ⊕ Z2 .
Thus H1 (T2 (N3 )) is isomorphic to Z2 ⊕ Z2 .
For g ≥ 2, the exact sequence (5.1) induces the five term exact sequence between
these groups:
H2 (Γ2 (Ng )) −→ H2 (Z2 ) −→ H1 (T2 (Ng ))Z2 −→ H1 (Γ2 (Ng )) −→ H1 (Z2 ) −→ 0,
where
H1 (T2 (Ng ))Z2 := H1 (T2 (Ng ))/ f m − m | m ∈ H1 (T2 (Ng )), f ∈ Z2 .
−1

For m ∈ H1 (T2 (Ng )) and f ∈ Z2 , f m := [f ′ m′ f ′ ] ∈ H1 (T2 (Ng )) for some
representative m′ ∈ T2 (Ng ) and f ′ ∈ Γ2 (Ng ). Since H2 (Z2 ) ∼
= H2 (RP ∞ ) = 0 and
∼
H1 (Z2 ) = Z2 , we have the short exact sequence:
0 −→ H1 (T2 (Ng ))Z2 −→ H1 (Γ2 (Ng )) −→ Z2 −→ 0.
(g)+(g)
Since Hirose and Sato [2] showed that H1 (Γ2 (Ng )) ∼
= Z23 2 , it is sufficient for the
proof of Theorem 1.2 when g ≥ 5 to prove that the action of Z2 ∼
= Γ2 (Ng )/T2 (Ng )
on the set of the first homology classes of generators for T2 (Ng ) is trivial.
By Theorem 1.1, T2 (Ng ) is generated by crosscap pushing maps along nonseparating two-sided simple loops for g ≥ 5. Let ψ(γ) = tγ1 t−1
γ2 be a crosscap
pushing map along a non-separating two-sided simple loop γ, where γ1 and γ2 are
images of boundary components of the regular neighborhood of γ in Ng−1 to Ng by
a blowup. The surface S obtained by cutting Ng along γ1 and γ2 is diffeomorphic
to a disjoint sum of Ng−3,2 and N1,2 . Since g − 3 ≥ 5 − 3 = 2, we can define
a Y-homeomorphism Y on the component of S. The Y-homeomorphism is not
a product of Dehn twists. Hence [Y ] is the nontrivial element in Z2 and clearly

18

R. KOBAYASHI AND G. OMORI

Y ψ(γ)Y −1 = ψ(γ) in Γ2 (Ng ), i.e. [Y ψ(γ)Y −1 ] = [ψ(γ)] in H1 (T2 (Ng )). Therefore
the action of Z2 on H1 (T2 (Ng )) is trivial and we have completed the proof of
Theorem 1.2.

Remark 5.1. When g = 3, we have the exact sequence
0 −→ H1 (T2 (N3 ))Z2 −→ H1 (Γ2 (N3 )) −→ Z2 −→ 0
by the argument in the proof of Theorem 1.2. Since H1 (Γ2 (N3 )) ∼
= H1 (Γ2 (2)) ∼
= Z42 ,
∼
we showed that the action of Z2 = Γ2 (N3 )/T2 (N3 ) on H1 (T2 (N3 )) is not trivial by
Theorem 1.2 when g = 3.
Acknowledgement:
The authors would like to express their gratitude to
Hisaaki Endo and Susumu Hirose, for his encouragement and helpful advices. The
authors also wish to thank Masatoshi Sato for his comments and helpful advices.
The second author was supported by JSPS KAKENHI Grant number 15J10066.
References
[1] D. R. J. Chillingworth, A finite set of generators for the homeotopy group of a non-orientable
surface, Math. Proc. Camb. Philos. Soc. 65 (1969), 409–430.
[2] S. Hirose, M. Sato, A minimal generating set of the level 2 mapping class group of a nonorientable surface, Math. Proc. Camb. Philos. Soc. 157 (2014), no. 2, 345–355.
[3] S. P. Humphries, Normal closures of powers of Dehn twists in mapping class groups, Glasgow
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[4] D. L. Johnson, Presentations of Groups, London Math. Soc. Stud. Texts 15 (1990).
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133.
[7] W. B. R. Lickorish, On the homeomorphisms of a non-orientable surface, Math. Proc. Camb.
Philos. Soc . 61 (1965), 61–64.
[8] J. D. McCarthy and U. Pinkall. Representing homology automorphisms of nonorientable
surfaces, Max Planck Inst. Preprint MPI/SFB 85-11, revised version written on 26 Feb 2004,
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[9] L. Paris and B. Szepietowski. A presentation for the mapping class group of a nonorientable
surface, arXiv:1308.5856v1 [math.GT], 2013.
[10] M. Stukow, The twist subgroup of the mapping class group of a nonorientable surface, Osaka
J. Math. 46 (2009), 717–738.
[11] B. Szepietowski. Crosscap slides and the level 2 mapping class group of a nonorientable
surface, Geom. Dedicata 160 (2012), 169–183.
[12] B. Szepietowski. A finite generating set for the level 2 mapping class group of a nonorientable
surface. Kodai Math. J. 36 (2013), 1–14.
(Ryoma Kobayashi) Department of General Education, Ishikawa National College
of Technology, Tsubata, Ishikawa, 929-0392, Japan
E-mail address: kobayashi ryoma@ishikawa-nct.ac.jp
(Genki Omori) Department of Mathematics, Tokyo Institute of Technology, Ohokayama, Meguro, Tokyo 152-8551, Japan
E-mail address: omori.g.aa@m.titech.ac.jp

