A twisted first homology group of the Goeritz group of 3-sphere

TL;DR

This paper computes the twisted first homology group of the genus-2 Goeritz group of the 3-sphere, providing new algebraic insights into the symmetries of Heegaard splittings.

Contribution

It is the first to explicitly determine the twisted first (co)homology group of the genus-2 Goeritz group of the 3-sphere.

Findings

Computed the twisted first homology group of the genus-2 Goeritz group.

Provided algebraic structure insights into the symmetries of genus-2 Heegaard splittings.

Enhanced understanding of the mapping class group actions on 3-sphere splittings.

Abstract

Given a genus-g Heegaard splitting of a 3-sphere, the genus-g Goeritz group is defined to be the group of the isotopy classes of orientation preserving homeomorphism of the 3-sphere that preserve the splitting. In this paper, we determine the twisted first (co)homology group of the genus-2 Goeritz group of 3-sphere.

Equations

$$H_2({\mathcal{M}}_g;\mathbb{Z})\cong \mathbb{Z}\text{if}g\ge 4.$$

$$H_1({\mathcal{M}}_g;{\mathcal{H}}_1({\Sigma }_g))\cong \mathbb{Z}/(2g-2)\mathbb{Z}\text{if}g\ge 2.$$

$$H_1({\mathcal{H}}_g;{\mathcal{H}}_1({\Sigma }_g))\cong \{\begin{array}{ll}\mathbb{Z}/(2g-2)\mathbb{Z}& \text{if}g\ge 4,\\ \mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/4\mathbb{Z}& \text{if}g=3,\\ (\mathbb{Z}/2\mathbb{Z}{)}^2& \text{if}g=2.\end{array}$$

$$H_1({\mathcal{E}}_2;{\mathcal{H}}_1({\Sigma }_2))\cong (\mathbb{Z}/2\mathbb{Z}{)}^2.$$

$$Z^1({\mathcal{E}}_g;{\mathcal{H}}_{\mathcal{A}})=\{d:{\mathcal{E}}_g\to {\mathcal{H}}_{\mathcal{A}};d(\varphi \psi )=d(\varphi )+\varphi d(\psi ),\varphi ,\psi \in {\mathcal{E}}_g\}.$$

$$\pi (u)(\varphi )=\varphi u-u$$

$$H^1({\mathcal{E}}_g;{\mathcal{H}}_{\mathcal{A}})={\mathcal{Z}}^1({\mathcal{E}}_g;{\mathcal{H}}_{\mathcal{A}})/\mathrm{Im}\pi$$

$$H_1({\Sigma }_2{)}_{{\mathcal{E}}_2}=0.$$

$$\cdots \to G_3\to G_2\to K\to G_1\to 0\text{and}$$

$$0\to H_1\to K\to H_2\to H_3\to \cdots$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$H_1\oplus H_2\stackrel{\simeq }{⟶}K:(h_1,h_2)\mapsto f_1(h_1)+g_2\circ {\Psi }^{-1}(h_2),$$

$$K\stackrel{\simeq }{⟶}{G}_1\oplus G_2:k\mapsto (g_1(k),{\Psi }^{-1}\circ f_2(k)).$$

$$(h_1,h_2)$$

$$H^1({\mathcal{E}}_2;{\mathcal{H}}_{\mathcal{A}})$$

$$d([\delta ]{)}_1-d([\alpha ]{)}_1=d([\gamma ]{)}_2-d([\beta ]{)}_2=d([\gamma ]{)}_3-d([\alpha ]{)}_3=d([\beta ]{)}_4-d([\delta ]{)}_4=0\}.$$

$$f(d)=(d([\delta ]{)}_1-d([\alpha ]{)}_1,d([\gamma ]{)}_2-d([\beta ]{)}_2,d([\gamma ]{)}_3-d([\alpha ]{)}_3,d([\beta ]{)}_4-d([\delta ]{)}_4).$$

$$\alpha v-v=$$

$$\beta v-v=$$

$$\gamma v-v=$$

$$\delta v-v=$$

$$f\circ \pi (v)=(v_2,-v_1+v_2,v_3-v_4,-v_3)$$

$$\mathrm{Ker}f\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$H^1({\mathcal{E}}_2;{\mathcal{H}}_{\mathcal{A}})={\mathcal{Z}}^1({\mathcal{E}}_2;{\mathcal{H}}_{\mathcal{A}})/\mathrm{Im}\pi \cong \mathrm{Ker}f.$$

$$d([\alpha ])={\omega }_{11}{x}_1+{\omega }_{12}{x}_2+{\omega }_{13}{y}_1+{\omega }_{14}{y}_2,$$

$$d([\beta ])={\omega }_{21}{x}_1+{\omega }_{22}{x}_2+{\omega }_{23}{y}_1+{\omega }_{24}{y}_2,$$

$$d([\gamma ])={\omega }_{31}{x}_1+{\omega }_{32}{x}_2+{\omega }_{33}{y}_1+{\omega }_{34}{y}_2,$$

$$d([\delta ])={\omega }_{41}{x}_1+{\omega }_{42}{x}_2+{\omega }_{43}{y}_1+{\omega }_{44}{y}_2.$$

$$H^1({\mathcal{E}}_2;{\mathcal{H}}_{\mathcal{A}})\cong \{d\in {\mathcal{Z}}^1({\mathcal{E}}_2;{\mathcal{H}}_{\mathcal{A}});{\omega }_{41}-{\omega }_{11}={\omega }_{32}-{\omega }_{22}={\omega }_{33}-{\omega }_{13}={\omega }_{24}-{\omega }_{44}=0\}.$$

$$\{\begin{array}{ll}{\omega }_{11}+{\omega }_{12}={\omega }_{31}+{\omega }_{32},& (1a)\\ {\omega }_{13}+{\omega }_{14}={\omega }_{33}+{\omega }_{34}.& (1b)\end{array}$$

$$d((\alpha \gamma {)}^2)$$

$$\{\begin{array}{ll}2{\omega }_{11}+{\omega }_{12}=2{\omega }_{41},& (2a)\\ -{\omega }_{11}+{\omega }_{12}=2{\omega }_{42},& (2b)\\ {\omega }_{13}+{\omega }_{14}=2{\omega }_{43},& (2c)\\ -{\omega }_{13}+2{\omega }_{14}=2{\omega }_{44},& (2d)\end{array}$$

$$d(\alpha \delta \alpha )$$

$$d(\delta )$$

$$\{\begin{array}{ll}2{\omega }_{21}=2{\omega }_{23}=0,& (3a)\\ 2{\omega }_{12}=2{\omega }_{22},& (3b)\\ 2{\omega }_{14}=2{\omega }_{24}.& (3c)\end{array}$$

Full text

1. Introduction

Mapping class group

Let $H_{g}$ be a 3-dimensional handlebody of genus $g$, and $\Sigma_{g}$ be the boundary surface $\partial H_{g}$. We denote by $\cal{M}_{g}$ the mapping class group of $\Sigma_{g}$, the group of isotopy classes of orientation preserving homeomorphisms of $\Sigma_{g}$. Dehn [2] proved that $\cal{M}_{g}$ is generated by finitely many Dehn twists. Furthermore Lickorish [19, 20] proved that $3g-1$ Dehn twists generate $\cal{M}_{g}$. Humphries [16] found that $2g+1$ Dehn twists generate $\cal{M}_{g}$.

 We denote by $\cal{H}_{g}$ the handlebody mapping class group, the subgroup of mapping class group $\cal{M}_{g}$ of boundary surface $\partial H_{g}$ defined by isotopy classes of those orientation preserving homeomorphisms of $\partial H_{g}$ which can be extended to homeomorphisms of $H_{g}$. It turns out that $\cal{H}_{g}$ can be identified with the group of isotopy classes of orientation preserving homeomorphisms of $H_{g}$. A finite presentation of the handlebody mapping class group $\cal{H}_{g}$ was obtained by Wajnryb [1].

Goeritz group

Let $H_{g}$ and $H_{g}^{*}$ be 3-dimensional handlebodies, and $M=H_{g}\cup H_{g}^{*}$ be a Heegaard splitting of a closed orientable $3$-manifold $M$. Let $\cal{M}_{g}$ be the mapping class group of the boundary surface $\partial H_{g}=\Sigma_{g}$. The group of mapping classes $[f]\in\cal{M}_{g}$ such that there is an orientation preserving self-homeomorphism $F$ of $(M,H_{g})$ satisfying $[F|_{\partial H_{g}}]=[f]$ is called the genus-$g$ Goeritz group of $M=H_{g}\cup H_{g}^{*}$. When a manifold $M$ admits a unique Heegaard splitting of genus $g$ up to isotopy, we can define the genus-$g$ Goeritz group of the manifold without mentioning a specific splitting. For example, the $3$-sphere, $S^{1}\times S^{2}$ and lens spaces are known to be such manifolds from [6], [4] and [5].

In studying Goeritz groups, finding their generating sets or presentations has been an interesting problem. However the generating sets or the presentation of those groups have been obtained only for a few manifolds with their splittings of small genera. A finite presentation of the genus-2 Goeritz group of 3-sphere was obtained [3]. In an arbitrary genus, first Powell [8] and then Hirose [17] claimed that they have found a finite generating set for the genus-$g$ Goeritz group of 3-sphere, though serious gaps in both arguments were found by Scharlemann. Establishing the existence of such generating sets appears to be an open problem.

In addition, finite presentations of the genus-2 Goeritz groups of each lens spaces $L(p,1)$ were obtained [12], other lens spaces were obtained [15] and the genus-$2$ Heegaard splittings of non-prime $3$-manifolds were obtained [14]. Recently a finite presentation of the genus-2 Goeritz group of $S^{1}\times S^{2}$ was obtained [13].

Homology of mapping class group

Computing homology of mapping class groups is interesting topic of studying mapping class groups. Harer [9] determined the second homology group of mapping class group $\cal{M}_{g}$:

[TABLE]

In fact, Harer proved a more general theorem for surfaces with multiple boundary components and arbitrarily many punctures.

In twisted case, Morita [18] determined the first homology group with coefficients in the first integral homology group of the surface:

[TABLE]

Recently Ishida and Sato [7] computed the twisted first homology groups of the handlebody mapping class group $\cal{H}_{g}$ with coefficients in the first integral homology group of the boundary surface $\Sigma_{g}$:

[TABLE]

Goeritz group of $S^{3}$

Let $H_{g}$ and $H_{g}^{*}$ be 3-dimensional handlebodies, and $S^{3}=H_{g}\cup H_{g}^{*}$ be the Heegaard splitting of the $3$-sphere $S^{3}$. Waldhausen [6] proved that a genus-$g$ Heegaard splitting of $S^{3}$ is unique up to isotopy. Let $\cal{M}_{g}$ be the mapping class group of the boundary surface $\partial H_{g}=\Sigma_{g}$. The group of mapping classes $[f]\in\cal{M}_{g}$ such that there is an orientation preserving self-homeomorphism $F$ of $(S^{3},H_{g})$ satisfying $[F|_{\partial H_{g}}]=[f]$ is denoted by $\cal{E}_{g}$. It is called the genus-$g$ Goeritz group of the $3$-sphere.

Twisted homology group of $\cal{E}_{2}$

In this paper, we compute the twisted first homology group of $\cal{E}_{2}$ with coefficients in the first integral homology group of the Heegaard surface $\Sigma_{2}$. The following is the main theorem in this paper.

Theorem 1.1**.**

[TABLE]

A finite presentation of the genus-2 Goeritz group of the 3-sphere was obtained from the works of [3]. For the higher genus Goeritz groups of the 3-sphere, it is conjectured that all of them are finitely presented however it is still known to be an open problem.

Let $\Sigma_{g}$ be a compact connected orientable surface of genus $g\geq 1$ and $\alpha_{1},\dots,\alpha_{g},\beta_{1},\dots,\beta_{g}$ be oriented simple closed curves as in Figure 1. We denote their homology classes in $H_{1}(\Sigma_{g})$ by $x_{1}=[\alpha_{1}],x_{2}=[\alpha_{2}],\dots,x_{g}=[\alpha_{g}],y_{1}=[\beta_{1}],y_{2}=[\beta_{2}],\dots,y_{g}=[\beta_{g}]$. The basis $\{x_{1},...,x_{g},y_{1},...,y_{g}\}$ of $H_{A}$ induces an isomorphism $H_{A}\simeq A^{2g}$. For $v\in A^{2g}$, we denote its projection to the $i$-th coordinate of $A^{2g}$ by $v_{i}$ for $i=1,2,...,2g$.

\begin{overpic}[width=312.9803pt]{s1} \put(15.0,32.0){$\alpha_{1}$} \put(37.0,36.0){$\alpha_{2}$} \put(83.0,32.0){$\alpha_{g}$} \put(15.0,17.0){$1$} \put(37.0,17.0){$2$} \put(83.0,17.0){$g$} \put(15.0,8.0){$\beta_{1}$} \put(37.0,8.0){$\beta_{2}$} \put(83.0,8.0){$\beta_{g}$} \put(0.0,-5.0){Figure 1 : Surface $\Sigma_{g}$ and simple closed curves $\alpha_{1},\dots,\alpha_{g},\beta_{1},\dots,\beta_{g}$.} \end{overpic}

Akbas gave following presentation for $\cal{E}_{2}$ in [3].

Theorem 1.2** ([3]).**

The group $\cal{E}_{2}$ has four generators $[\alpha],[\beta],[\gamma]$ and $[\delta]$, and the following relations:

• (P1)

$[\alpha]^{2}=[\beta]^{2}=[\delta]^{3}=[\alpha\gamma]^{2}=0$.

• (P2)

$[\alpha\delta\alpha]=[\delta]$* and $[\alpha\beta\alpha]=[\beta]$.*

• (P3)

$[\gamma\beta\gamma]=[\alpha\beta]$* and $[\delta]=[\gamma\delta^{2}\gamma]$.*

\begin{overpic}[width=312.9803pt]{s2} \put(-5.0,75.0){(i)\ \ $\alpha$:} \put(48.0,68.0){$\pi$} \put(48.0,75.0){(ii)\ \ $\beta$:} \put(98.0,68.0){$\pi$} \put(-5.0,38.0){(iii)\ \ $\gamma$:} \put(15.0,42.0){$\pi$} \put(48.0,38.0){(iv)\ \ $\delta$:} \put(65.0,45.0){$\frac{2\pi}{3}$} \put(22.0,-5.0){Figure 2 : Generators of $\cal{E}_{2}$.} \end{overpic}

We define $\delta$ as follows. Consider the genus-two handlebody as a regular neighborhood of a sphere, centered at the origin, with three holes. The homeomorphism $\delta$ is a ${2\pi}/{3}$ rotation of the handlebody about the vertical $z$-axis. See Figure 2. Scharlemann [11] showed that the group $\cal{E}_{2}$ is generated by isotopy classes $[\alpha],[\beta],[\gamma]$ and $[\delta]$. Correspondence of homology classes of (iv) and the others are as follows:

\begin{overpic}[width=199.16928pt]{s3} \put(14.0,35.0){$x_{1}$} \put(14.0,40.5){$<$} \put(40.0,42.2){\rotatebox{-20.0}{>}} \put(38.5,37.0){$x_{1}-x_{2}$} \put(80.0,41.2){$>$} \put(80.0,36.0){$x_{2}$} \put(40.0,17.2){\rotatebox{-30.0}{>}} \put(30.0,11.0){$y_{1}+y_{2}$} \put(32.5,60.0){\rotatebox{-64.0}{<}} \put(37.5,60.0){$y_{1}$} \put(61.0,55.0){\rotatebox{70.0}{<}} \put(57.0,60.0){$y_{2}$} \end{overpic}

2. Twisted cohomology group of $\cal{E}_{2}$

We denote by $A$ the ring $\mathbb{Z}$ or $\mathbb{Z}/n\mathbb{Z}$ for an integer $n\geq 2$, and set $H_{A}=H_{1}(\Sigma_{g};A)$. For a group $G$ and a left $G$-module $N$, a map $d:G\to N$ is called a crossed homomorphism if it satisfies $d(gg^{\prime})=d(g)+gd(g^{\prime})$ for $g,g^{\prime}\in G$. Now let $Z^{1}(\cal{E}_{g};H_{A})$ be the set of all crossed homomorphisms $d:\cal{E}_{g}\to H_{A}$. Namely

[TABLE]

Let $\pi:H_{A}\to Z^{1}(\cal{E}_{g};H_{A})$ be the homomorphism defined by

[TABLE]

for $u\in H_{A}$. Then as is well known we have

[TABLE]

(cf. K.S. Brown [10]).

We consider the case $g=2$. Then we have the homomorphism $\cal{E}_{2}\to$ Aut$(H_{1}(\Sigma_{2};\mathbb{Z}))$ induced by the action of the group $\cal{E}_{2}$ on $H_{1}(\Sigma_{2};\mathbb{Z})$. The action of $\alpha,\beta,\gamma,$ and $\delta$ is as follows:

• $\alpha_{*}$:

$\alpha_{*}(x_{i})=-x_{i}$ and $\alpha_{*}(y_{i})=-y_{i}$ ($i=1,2$).

• $\beta_{*}$:

$\beta_{*}(x_{1})=x_{1}$, $\beta_{*}(x_{2})=-x_{2}$, $\beta_{*}(y_{1})=y_{1}$, $\beta_{*}(y_{2})=-y_{2}$.

• $\gamma_{*}$:

$\gamma_{*}(x_{1})=-x_{2}$, $\gamma_{*}(x_{2})=-x_{1}$,$\gamma_{*}(y_{1})=-y_{2}$, $\gamma_{*}(y_{2})=-y_{1}$.

• $\delta_{*}$:

$\delta_{*}(x_{1})=-x_{1}+x_{2}$, $\delta_{*}(x_{2})=-x_{1}$, $\delta_{*}(y_{1})=y_{2}$, $\delta_{*}(y_{2})=-y_{1}-y_{2}$.

For a group $G$ and a left $G$-module $N$, the coinvariant $N_{G}$ is quotient module of $N$ by the subgroup $\{gn-n|g\in G,n\in N\}$.

Lemma 2.1**.**

[TABLE]

Proof.

Since we have $\alpha_{*}\delta_{*}^{2}(-x_{2})=x_{1}-x_{2}$ and $\gamma_{*}(x_{1})=-x_{2}$, we obtain $x_{1}=x_{2}=0\in H_{1}(\Sigma_{2})_{\mathcal{E}_{2}}$. And we have $\gamma_{*}\delta_{*}^{2}(y_{1})=y_{1}+y_{2}$ and $\gamma_{*}(y_{1})=-y_{2}$. Hence we also obtain $y_{1}=y_{2}=0\in H_{1}(\Sigma_{2})_{\mathcal{E}_{2}}$. ∎

Lemma 2.2**.**

Let $G_{i}$, $H_{i}$ and $K$ be G-modules $(i=1,2,3)$, and let

[TABLE]

be exact sequences. If $G_{2}\to K\to H_{2}$ is an isomorphism, then we have $H_{1}\to K\to G_{1}$ is isomorphism.

Proof.

Now we have the following diagram. Set $\Phi=g_{1}\circ f_{1}$ and $\Psi=f_{2}\circ g_{2}$.

[TABLE]

Note that $g_{2}$ is injective and $f_{2}$ is surjective if $G_{2}\to K\to H_{2}$ is an isomorphism. Thus we have $g_{3}=0$ and $f_{3}=0$. Since $\Psi$ is isomorphism, we have $(\Psi^{-1}\circ f_{2})\circ g_{2}=id_{G_{2}}$ and $f_{2}\circ(g_{2}\circ\Psi^{-1})=id_{H_{2}}$. Hence those exact sequences are split and we have

[TABLE]

A composition map $H_{1}\oplus H_{2}\to K\to G_{1}\oplus G_{2}$ is

[TABLE]

Hence $\Psi$ is an isomorphism. ∎

Lemma 2.3**.**

In the case $g=2$, we have

[TABLE]

Proof.

Let $f:Z^{1}(\cal{E}_{2};H_{A})\to A^{4}$ be a homomorphism defined by

[TABLE]

Since we have

[TABLE]

the composition map $f\circ\pi\hskip 2.84526pt:\hskip 2.84526ptH_{A}\to A^{4}$ is written as

[TABLE]

for $v\in H_{A}$. This map is an isomorphism. We have the following diagram.

[TABLE]

By Lemma $2.2$, we have

[TABLE]

∎

The group $\cal{E}_{2}$ is generated by $[\alpha],[\beta],[\gamma],$ and $[\delta]$. Therefore, all crossed homomorphisms $d:\cal{E}_{2}\to H_{A}$ are determined by the values $d([\alpha]),d([\beta]),d([\gamma])$ and $d([\delta])$. If $d\in Z^{1}(\cal{E}_{2},H_{A})$, we can set

[TABLE]

Then we have

[TABLE]

3. relations of $\omega_{ij}$

In this section we shall consider the case $g=2$. We denote $d([\phi])$ for $\phi\in\cal{E}_{2}$ simply by $d(\phi)$.

Lemma 3.1**.**

We have relations:

[TABLE]

Proof.

By the relations $(\alpha\gamma)^{2}=0$ in (P1), we have $d((\alpha\gamma)^{2})=0$. The equation

[TABLE]

holds. Hence, we obtain $(1a)$ and $(1b)$. ∎

Lemma 3.2**.**

We have relations:

[TABLE]

Proof.

By the relations $\alpha\delta\alpha=\delta$ in (P2), we have $d(\alpha\delta\alpha)=d(\delta)$ and

[TABLE]

Comparing $d(\alpha\delta\alpha)$ and $d(\delta)$, we obtain $(2a)-(2d)$. ∎

Lemma 3.3**.**

We have relations:

[TABLE]

Proof.

By the relation $\alpha\beta\alpha=\beta$ in (P2), we have $d(\alpha\beta\alpha)=d(\beta)$ and

[TABLE]

Comparing $d(\alpha\beta\alpha)$ and $d(\beta)$, we obtain $(3a)-(3c)$. ∎

Lemma 3.4**.**

We have relations:

[TABLE]

Proof.

By the relation $\gamma\beta\gamma=\alpha\beta$ in (P3), we have $d(\gamma\beta\gamma)=d(\alpha\gamma)$ and

[TABLE]

Comparing $d(\gamma\beta\gamma)$ and $d(\alpha\beta)$, we obtain $(4a)-(4d)$. ∎

Lemma 3.5**.**

We have relations:

[TABLE]

Proof.

By the relation $\gamma\delta^{2}\gamma=\delta$ in (P3), we have $d(\gamma\delta^{2}\gamma)=d(\delta)$ and

[TABLE]

Comparing $d(\gamma\delta^{2}\gamma)$ and $d(\delta)$, we obtain $(5a)$ and $(5b)$. ∎

By Lemma 3.1, 3.2, 3.3, 3.4 and 3.5, we obtain the following equations.

      $\begin{cases}\omega_{11}+\omega_{12}=\omega_{31}+\omega_{32},&(1a)\\ \omega_{13}+\omega_{14}=\omega_{33}+\omega_{34},&(1b)\end{cases}$       $\begin{cases}2\omega_{11}+\hskip 2.84526pt\hskip 2.84526pt\omega_{12}=2\omega_{41},&(2a)\\ \hskip 2.84526pt-\omega_{11}+\omega_{12}=2\omega_{42},&(2b)\\ \hskip 2.84526pt\hskip 2.84526pt\omega_{13}+\hskip 2.84526pt\hskip 2.84526pt\omega_{14}=2\omega_{43},&(2c)\\ -\omega_{13}+2\omega_{14}=2\omega_{44},&(2d)\end{cases}$

      $\begin{cases}2\omega_{21}=2\omega_{23}=0,&(3a)\\ 2\omega_{12}=2\omega_{22},&(3b)\\ 2\omega_{14}=2\omega_{24},&(3c)\end{cases}$ $\hskip 14.22636pt\hskip 14.22636pt\hskip 14.22636pt\begin{cases}\omega_{31}-\omega_{32}-\omega_{22}=\omega_{11}+\omega_{21},&(4a)\\ \omega_{31}+\omega_{32}-\omega_{21}=\omega_{12}-\omega_{22},&(4b)\\ \omega_{33}-\omega_{34}-\omega_{24}=\omega_{13}+\omega_{23},&(4b)\\ \omega_{34}+\omega_{33}-\omega_{23}=\omega_{14}-\omega_{24},&(4b)\end{cases}$

      $\begin{cases}2\omega_{31}+\omega_{32}=2\omega_{41}+\omega_{42},&(5a)\\ \hskip 34.14322pt\omega_{33}=\hskip 2.84526pt\hskip 2.84526pt\omega_{43}.&(5b)\end{cases}$

4. Calculation of cohomology

In this section we prove that $H^{1}(\mathcal{E}_{2};H_{A})\cong\operatorname{Hom}((\mathbb{Z}/2\mathbb{Z})^{2},A)$. The universal coefficient theorem implies $H_{1}(\cal{E}_{2};H_{1}(\Sigma_{2}))\cong(\mathbb{Z}/2\mathbb{Z})^{2}$. To determine the twisted first cohomology group of $\cal{E}_{2}$, we solve equations $(1a)-(5b)$ and the condition $d\in\operatorname{Ker}f$, i.e.

[TABLE]

Lemma 4.1**.**

We have a relation of $\operatorname{Ker}f$ :

[TABLE]

Proof.

Using $(2a)$ and $\omega_{11}=\omega_{41}$ by $(*)$, we obtain

[TABLE]

∎

Lemma 4.2**.**

The elements $\omega_{21}$, $\omega_{22}$, $\omega_{31}$, $\omega_{32}$ and $\omega_{42}$ have order $2$ and

[TABLE]

Proof.

By (2a) and (2b), we have $\omega_{11}+2\omega_{12}=2\omega_{41}+2\omega_{42}$. By $(5a)$ we have $2\omega_{41}+2\omega_{42}=2\omega_{31}+\omega_{32}+\omega_{42}$. Using these two equations and $(1a)$, we have $\omega_{12}=\omega_{31}+\omega_{42}.$ Since $\omega_{12}=0$, the equation

[TABLE]

holds. Using $(2b)$ and Lemma 4.1, we obtain

[TABLE]

The equation

[TABLE]

holds. So we obtain

[TABLE]

We have

[TABLE]

Here, $(**)$ is the equation $\omega_{11}=2\omega_{22}$. This equation is obtained as follows:

[TABLE]

By $(3b)$ and Lemma4.1, we have $-2\omega_{22}=0$. Hence we obtain

[TABLE]

∎

From the results of Lemma 4.2 and (4.2.2), we have

[TABLE]

Lemma 4.3**.**

We have a relation:

[TABLE]

Proof.

By $(5b)$ and $(*)$, we obtain

[TABLE]

Using this equation and $(1b)$, we have $\omega_{14}=\omega_{34}$. So the equation

[TABLE]

holds. The equation

[TABLE]

holds. Hence we have

[TABLE]

So we obtain

[TABLE]

Since the equation $2\omega_{44}=\omega_{14}$ holds, we complete the proof of Lemma 4.3. ∎

Lemma 4.4**.**

The elements $\omega_{23}$, $\omega_{24}$ and $\omega_{44}$ have order $2$ and are equal to each other:

[TABLE]

Proof.

By (3a), the element $\omega_{23}$ have order $2$. Since the equation

[TABLE]

holds, we obtain

[TABLE]

∎

Proof of Theorem 1.1.

From Lemmas 3,7, 4.1, 4.2, 4.3 and 4.4, we have

[TABLE]

where $-\omega_{22}=\omega_{42}$ and $2\omega_{22}=2\omega_{23}=0$. Hence it follows

[TABLE]

So we obtain

[TABLE]

This isomorphism follows from the short exact sequence

[TABLE]

since we have $H_{0}(\cal{E}_{2};H_{1}(\Sigma_{2}))=H_{1}(\Sigma_{2})_{\cal{E}_{2}}=0$ by Lemma 2.1. we complete the proof of Theorem 1.1.