Cohomology of $BPU_n$ and rings of invariants of Weyl groups

TL;DR

This paper computes the integral cohomology groups and ring structure of the classifying space of the projective unitary group $PU_n$ up to certain dimensions, using spectral sequences and Weyl group invariants.

Contribution

It provides explicit calculations of the cohomology and ring structure of $BPU_n$, extending previous knowledge with detailed low-dimensional results.

Findings

Computed $H^*(BPU_n;\mathbb{Z})$ in dimensions up to 14.

Determined the ring of invariants $H^*(BT_{PU_n})^W$ up to dimension 12.

Established the ring structure of $H^*(BPU_n;\mathbb{Z})$ in dimensions up to 13.

Abstract

Let $PU_n$ denote the projective unitary group of rank $n$ and $BPU_n$ be its classifying space, for $n>1$. Using the Serre spectral sequence associated to the fibration $BU_n\to BPU_n\to K(\mathbb{Z},3)$, we compute the integral cohomology group of $BPU_n$ in dimensions $\leq 14$. In addition, we determine the ring structure of $H^*(BPU_n;\mathbb{Z})$ up to dimension $13$ by computing the ring of invariants $H^*(BT_{PU_n})^W$ of the Weyl group action in dimensions $\leq 12$.