On a question of Serre

TL;DR

This paper investigates the kernel of a homology map induced by boundary attachment in the Borel-Serre compactification of hyperbolic 3-space quotients, providing a cusp-by-cusp decomposition using local topology.

Contribution

It offers a local topological analysis to explicitly decompose the kernel of the homology map into cusp-related components, addressing Serre's question.

Findings

Kernel decomposed into cusp-associated parts

Provides explicit local topological description

Addresses Serre's original question

Abstract

Consider the Borel-Serre compactification of the quotient of hyperbolic 3-space by a finite index subgroup in a Bianchi group, and in particular the following question which Serre posed on page 514 of the quoted article. Consider the map alpha induced on homology when attaching the boundary into the Borel-Serre compactification. How can one determine the kernel of alpha (in degree 1)? Serre used a global topological argument and obtained the rank of the kernel of alpha. In the quoted article, Serre did add the question what submodule precisely this kernel is. Through a local topological study, we can decompose the kernel of alpha into its parts associated to each cusp.