Third homology of perfect central extensions

Abstract

For a central perfect extension of groups $A\rightarrowtail G\twoheadrightarrow Q$, first we study the natural image of $H_3(A,\mathbb{Z})$ in $H_3(G,\mathbb{Z})$. As a particular case, we show that if the extension is universal this image is 2-torsion. Moreover when the plus-construction of the classifying space of $Q$ is an $H$-space, we also study the kernel of the surjective homomorphism $H_3(G,\mathbb{Z})\to H_3(Q,\mathbb{Z})$.

20 pages