# Functorality of universal central extension

I'm looking a reference (or quick proof) for the following fact, which doesn't appear in the standard sources I've consulted (for instance, Milnor's book on algebraic k-theory).

Let $\phi : G \rightarrow G'$ be a homomorphism between perfect groups. Set $A = H_2(G)$ and $A' = H_2(G')$, and let $$1 \longrightarrow A \longrightarrow \tilde{G} \longrightarrow G \longrightarrow 1$$ and $$1 \longrightarrow A' \longrightarrow \tilde{G}' \longrightarrow G' \longrightarrow 1$$ be the universal central extensions. Then there exists a unique homomorphism $\tilde{\phi} : \tilde{G} \rightarrow \tilde{G}'$ lifting $\phi$. Also, the restriction $A \rightarrow A'$ of $\tilde{\phi}$ to $A$ is the map $A \rightarrow A'$ induced by $\phi$.

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My case of interest is universal central extension of the fundamental group of a compact Riemann surface:does it exist as claimed by Atiyah-Bott p559 "Yang-Mills Equations over Riemann Surfaces" Phil Traans Royal Soc 1982 pp523-615 ? And the operations following this remark need elaboration. S.Srinivas Rau –  rauindia Nov 18 '13 at 11:37
@rauindia If you have a new question, you should ask it as a separate question by clicking the "Ask Question" button, rather than as an answer or comment for another question. –  Ben Webster Nov 18 '13 at 16:03

1) A concise exposition of central extensions is given by Steinberg in Section 7 of his 1967-68 Yale lectures on Chevalley groups, distributed in mimeographed form and currently linked on his UCLA homepage here. Item (ix) at the bottom of page 76 is what you need. The proof is elementary, and standard, but note an obvious typo on page 77: the first handwritten symbol $\psi$ should have been $\pi$.
2) An even more concise account can be found in the important 1968 paper by Calvin Moore here. See his statement in the middle of page 10. As indicated by the title Group extensions of $p$-adic and adelic linear groups, Moore came to these ideas from a rather different direction. But his approach and Steinberg's soon merged in the work of Matsumoto and others on the Congruence Subgroup Property in the case of Chevalley (split) groups over number fields.