Universal Central Extension of pi(X), X a compact Riemann surface of genus>1 Does a universal central extension exist for the fundamental group of a Compact Riemann Surface of genus1? Please give a detailed explanation.I am unable to justify the statements in Atiyah-Bott Phil Trans Roy Soc 1982 p559 (apparently it exists and some quotient operations are performed)  
 A: A compact Riemann surface of genus one (with a chosen basepoint) has fundamental group $\mathbb{Z}^2$.  A group has a universal central extension if and only if it is perfect, and $\mathbb{Z}^2$ is not perfect.
A: Well, the OP is right: Atiyah and Bott indeed use the term "universal central extension" for the fundamental group  of a compact surface $\Sigma_g$ of genus $g\geq 1$ -- which is incorrect since that group is not perfect. What they mean is that there is in this case a canonical central extension by $\mathbb{Z}$: indeed these extensions are parameterized by $H^2(\pi _1(\Sigma _g),\mathbb{Z})$; since $\Sigma _g$  is a $K(\Pi ,1)$, this is canonically isomorphic to $H^2(\Sigma _g, \mathbb{Z})=\mathbb{Z}$, and they consider the extension associated to the generator of this group. They describe  explicitly this extension
$$0\rightarrow \mathbb{Z}\rightarrow \tilde{\pi }_g\rightarrow \pi _1(\Sigma _g)\rightarrow 1 $$as follows: $\tilde{\pi }_g $ is the largest quotient of the free group $F_{2g}$, 
with generators  $a_1,\ldots ,a_g,b_1,\ldots ,b_g$, such that  the element $[a_1,b_1]\ldots [a_g,b_g]$ is central.  
