I asked this question on Mathematics Stack Exchange some months ago but I got no answer.


Suppose one has an orientable compact surface $S$ of genus $g\ge 2$, $x\in S$, and $G=\pi(S,x)$ the fundamental group. There is a well-known description of the group with generators and relations as $$G=\langle a\_1,b\_1,\dots, a\_g,b\_g \ \mid \ [a\_1,b\_1]\cdots[a\_g,b\_g]=1\rangle$$
where $[a,b]=aba^{-1}b^{-1}$ as usual.


Now, it is know that the homology $H\_2(S,\mathbb{Z})\cong \mathbb{Z}$ and that $$H\_2(G,\mathbb{Z})\cong H\_2(S,\mathbb{Z})$$
using group cohomology.


On the other hand the group cohomology can be descrived using the so called bar resolution, whose elements $z$ are in $\mathbb{Z}[G]\otimes \mathbb{Z}[G]$ such that $\delta\_2(z)=0$, where $\delta\_2(g\otimes h)= g+h-gh$, where $\delta\_2$ goes to $\mathbb{Z}[G]$ (and then quotienting by the image of a $\delta\_3$ analogous to $\delta\_2$).


My question is: what is the generator of $H\_2(G,\mathbb{Z})$ expressed as an element in $\mathbb{Z}[G]\otimes \mathbb{Z}[G]$ in terms of the generators $a\_i$, $b\_i$, $i=1,\dots,g$ given above?


I am studying group cohomology using K.S. Brown book "Cohomology of groups".



A description of the fundamental class of the group cohomology of the fundamental group of an orientable surface of genus $g$
For $1\le i\le g$, consider the following elements of the second degree of the bar complex:
\begin{align\*}
\alpha\_i &= \left(\prod\_{j < i} [a\_i,b\_i]\right)\otimes a\_i\\
\beta\_i &= \left(\prod\_{j < i} [a\_i,b\_i]\right)a\_i\otimes b\_i\\
\gamma\_i &= \left(\prod\_{j < i} [a\_i,b\_i]\right)a\_ib\_i\otimes a\_i^{-1}\\
\delta\_i &= \left(\prod\_{j < i} [a\_i,b\_i]\right)a\_ib\_ia\_i^{-1}\otimes b\_i^{-1}\\
\epsilon\_i &= a\_i\otimes a\_i^{-1}\\
\epsilon\_i' &= b\_i\otimes b\_i^{-1}\\
\zeta &= e\otimes e
\end{align\*}
Then the definition of the bar differential gives
\begin{align\*}
\mathrm d\alpha\_i &= \left(\prod\_{j < i} [a\_i,b\_i]\right) + a\_i - \left(\prod\_{j < i} [a\_i,b\_i]\right) a\_i\\
\mathrm d\beta\_i &= \left(\prod\_{j < i} [a\_i,b\_i]\right)a\_i + b\_i - \left(\prod\_{j < i} [a\_i,b\_i]\right) a\_ib\_i\\
\mathrm d\gamma\_i &= \left(\prod\_{j < i} [a\_i,b\_i]\right)a\_ib\_i + a\_i^{-1} - \left(\prod\_{j < i} [a\_i,b\_i]\right) a\_ib\_ia\_i^{-1}\\
\mathrm d\delta\_i &= \left(\prod\_{j < i} [a\_i,b\_i]\right)a\_ib\_ia\_i^{-1} + b\_i^{-1} - \left(\prod\_{j \le i} [a\_i,b\_i]\right)\\
\mathrm d\epsilon\_i &= a\_i + a\_i^{-1} - e\\
\mathrm d\epsilon\_i' &= b\_i + b\_i^{-1} - e\\
\mathrm d\zeta &= e + e - e = e\\
\mathrm d\big(\underbrace{\alpha\_i + \beta\_i + \gamma\_i + \delta\_i - \epsilon\_i - \epsilon\_i' - 2\zeta}\_{\xi\_i}\big) &= \left(\prod\_{j < i} [a\_i,b\_i]\right) - \left(\prod\_{j \le i} [a\_i,b\_i]\right)
\end{align\*}
It follows that $\xi = \sum\_{i=1}^g \xi\_i$ satisfies
\[
\mathrm d\xi = \sum\_{i=1}^g \left(\prod\_{j < i} [a\_i,b\_i]\right) - \left(\prod\_{j \le i} [a\_i,b\_i]\right) = e - \prod\_{i=1}^g [a\_i,b\_i] = 0
\]
so that $[\xi]\in H\_2(G;\mathbb Z)$ defines a homology class. To check that it is a generator, one can reduce to the case $g = 1$ using the map $a\_i\mapsto \delta\_{i,1}a\_1,b\_i\mapsto \delta\_{i,1}b\_1$ and use the commutative ring structure on the group homology of *abelian* groups.


This bar complex representative is essentially given by triangulating the $2g$-gon $2$-cell of the standard CW structure on $S$.


