Some torsion classes in the Chow ring and cohomology of $BPG{L}_n$

Abstract

In the integral cohomology ring of the classifying space of the projective linear group $PG{L}_n$ (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 $k\ge 0$. 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 $PG{L}_n$, 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.

35 pages, to appear in the Journal of the London Mathematical Society