On the cycle map of a finite group

TL;DR

This paper constructs a finite group of order p^{p+3} demonstrating that the mod p cycle map from the Chow ring to cohomology can fail to be injective, revealing new complexities in algebraic topology.

Contribution

It provides the first known example of a finite group where the mod p cycle map is not injective, highlighting limitations of existing cohomological tools.

Findings

Existence of a finite group with non-injective cycle map

The order of the constructed group is p^{p+3}

Shows limitations of the cycle map in algebraic topology

Abstract

Let p be an odd prime number. We show that there exists a finite group of order p^{p+3} whose the mod p cycle map from the mod p Chow ring of its classifying space to its ordinary mod p cohomology is not injective.