# Some torsion classes in the Chow ring and cohomology of $BPGL_n$

## TL;DR

This paper identifies specific torsion classes in the cohomology and Chow ring of the classifying space of PGL_n, revealing new torsion phenomena and their relation via the cycle class map.

## Contribution

It discovers explicit p-torsion classes in the cohomology and Chow ring of B PGL_n and relates them through the cycle class map, advancing understanding of their algebraic structure.

## Key findings

- Existence of p-torsion classes y_{p,k} in cohomology of B PGL_n.
- Construction of p-torsion classes ρ_{p,k} in the Chow ring mapping to y_{p,k}.
- Implications for Chern subrings and torsion phenomena in algebraic topology.

## Abstract

In the integral cohomology ring of the classifying space of the projective linear group $PGL_n$ (over $\mathbb{C}$), we find a collection of $p$-torsions $y_{p,k}$ of degree $2(p^{k+1}+1)$ for any odd prime divisor $p$ of $n$, and $k\geq 0$. If in addition, $p^2\nmid n$, there are $p$-torsion classes $\rho_{p,k}$ of degree $p^{k+1}+1$ in the Chow ring of the classifying stack of $PGL_n$, such that the cycle class map takes $\rho_{p,k}$ to $y_{p,k}$. We present an application of the above classes regarding Chern subrings.