Further counterexamples to the integral Hodge conjecture

TL;DR

This paper extends known counterexamples to the integral Hodge conjecture and related conjectures for certain classifying spaces, demonstrating their failure in broader contexts and for all primes.

Contribution

It proves a conjecture of Ben Antieau, generalizing counterexamples to the integral Hodge conjecture for a family of groups and extending these to all primes in finite field cases.

Findings

Counterexamples to the integral Hodge conjecture are extended to a broader family of groups.

The results apply to all primes in the context of finite fields.

The work confirms the failure of the integral $ ext{l}$-adic Tate conjecture in new cases.

Abstract

We study the integral Hodge conjecture in complex codimension $2$ and $3$ for approximations to the classifying space of groups of type A. In degree two, we prove a conjecture of Ben Antieau, extending his two counterexamples to a general family of groups. In particular, when reduced to a finite field, these spaces extend Antieau's counterexamples to the integral $\ell$-adic Tate conjecture from the cases $\ell = 2, 3$ to all primes $\ell$.