The $p$-primary subgroups of the cohomology of $BPU_n$ in dimensions less than $2p+5$

TL;DR

This paper provides a detailed description of the p-primary subgroups of the integral cohomology of the classifying space of the projective unitary group, extending previous results to dimensions less than 2p+5, and shows these subgroups are trivial in specific dimensions.

Contribution

It offers a complete description of the p-primary subgroups of H^s(BPU_n;Z) for s<2p+5, including new results on triviality at certain dimensions.

Findings

p-primary subgroups are trivial at s=2p+3 and s=2p+4

extends previous cohomology descriptions to lower dimensions

provides explicit structure for cohomology in specified range

Abstract

Let $PU_n$ the projective unitary group of rank $n$ and $BPU_n$ its classifying space. For an odd prime $p$, we extend previous results to a compete description of $H^s(BPU_n;\mathbb{Z})_{(p)}$ for $s<2p+5$ by showing that the $p$-primary subgroups of $H^s(BPU_n;\mathbb{Z})$ are trivial for $s = 2p+3$ and $s = 2p+4$.