A description of the fundamental class of the group cohomology of the fundamental group of an orientable surface of genus $g$ 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: 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 $S$.
