The cohomology of the classifying space of $PU(4)$

TL;DR

This paper computes the integral cohomology ring of the classifying space of the projective unitary group $PU(4)$ and analyzes its mod 2 cohomology structure, advancing understanding of its topological properties.

Contribution

It provides the first complete description of the integral cohomology ring of $BPU(4)$ and details the Steenrod algebra structure of its mod 2 cohomology.

Findings

Integral cohomology ring of $BPU(4)$ determined

Steenrod algebra structure of mod 2 cohomology analyzed

Enhanced understanding of the topology of $PU(4)$ classifying space

Abstract

Let $BPU(n)$ be the classifying space of the projective unitary group $PU(n)$. We determine the integral cohomology ring of $BPU(4)$, and the Steenrod algebra structure of its mod $2$ cohomology.