The ranks of homology of complexes of projective modules over finite groups

TL;DR

This paper extends counterexamples to a conjecture about the rank of homology in free complexes from specific cases to all finite groups, broadening the understanding of homological properties in algebraic topology.

Contribution

It generalizes counterexamples to the algebraic version of Gunnar Carlsson's conjecture across all finite groups, demonstrating the conjecture's limitations.

Findings

Counterexamples can be extended to any finite group.

The algebraic version of Carlsson's conjecture does not hold universally.

The results highlight the complexity of homology ranks in projective module complexes.

Abstract

We show that counterexamples of Iyengar and Walker to the algebraic version of Gunnar Carlsson's conjecture on the rank of the homology of a free complex can be extended to examples over any finite group with many choices of the complex.