Representation theory and the cycle map of a classifying space

TL;DR

This paper investigates the structure of the 4th integral cohomology of a specific classifying space, revealing it as a proper subgroup and providing new counterexamples to longstanding conjectures in algebraic geometry.

Contribution

It computes the Chern subgroup of the cohomology and demonstrates its properness, offering novel counterexamples to the integral Hodge and Tate conjectures.

Findings

The Chern subgroup is a proper subgroup of the cohomology group.

Provides new counterexamples to the integral Hodge conjecture.

Provides new counterexamples to the Tate conjecture.

Abstract

We compute the Chern subgroup of the 4-th integral cohomology group of a certain classifying space and show that it is a proper subgroup. Such a classifying space gives us new counterexamples for the integral Hodge and Tate conjectures modulo torsion.