Third homology of perfect central extensions

TL;DR

This paper investigates the third homology groups in perfect central extensions of groups, revealing torsion properties and kernel structures under specific topological conditions.

Contribution

It provides new insights into the image and kernel of third homology groups in perfect central extensions, especially for universal extensions and when the classifying space is an H-space.

Findings

Image of H_3(A,Z) in H_3(G,Z) is 2-torsion for universal extensions

Kernel of H_3(G,Z) to H_3(Q,Z) studied under H-space conditions

Enhanced understanding of homology in 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 $H_3(G,\mathbb{Z}) \to H_3(Q, \mathbb{Z})$.