Third homology of perfect central extensions

Abstract

For a central perfect extension of groups $A \rightarrowtail G \twoheadrightarrow Q$, we study the maps $H_3(A,\mathbb{Z}) \to H_3(G, \mathbb{Z})$ and $H_3(G, \mathbb{Z}) \to H_3(Q, \mathbb{Z})$ provided that $A\subseteq G'$. First we show that the image of $H_3(A, \mathbb{Z})\to H_3(G, \mathbb{Z})/\rho_\ast(A\otimes_\mathbb{Z} H_2(G, \mathbb{Z}))$ is $2$-torsion where $\rho: A \times G \to G$ is the usual product map. When $BQ^+$ is an $H$-space, we also study the kernel of the surjective homomorphism $H_3(G, \mathbb{Z}) \to H_3(Q, \mathbb{Z})$.