A formula for the mod $p$ cohomology of $BPU(p)$
Feifei Fan
TL;DR
The paper delivers a topological, odd-prime mod $p$ refinement of the classical integral formula for $H^*(BPU(p))$ by exploiting restriction maps to a maximal torus and to a key non-toral elementary abelian subgroup $\Gamma$. Central to the approach is a detailed description of how $H^*(BPU(p); \mathbb{F}_p)$ sits inside the restriction image $H^*(BU(p); \mathbb{F}_p) \times H^*(B\Gamma; \mathbb{F}_p)$ and how the SL$_2(\mathbb{F}_p)$-invariants control this image, together with explicit generators and relations involving $K_p$, $\delta$, and $c_1$. The main result presents $H^*(BPU(p); \mathbb{F}_p)$ as a quotient of $L_p \otimes Q_p$ by a specific ideal, mirroring Vistoli's integral structure, and a subsequent, simpler topological proof of Vistoli's formula is given via a parallel restriction-injection argument. These findings illuminate the mod $p$ cohomology landscape of $BPU(p)$ and provide tools for studying $H^*(BPU(n))$ when $p|n$ in broader contexts.
Abstract
We study the mod $p$ cohomology ring of the classifying space $BPU(p)$ of the projective unitary group $PU(p)$, when $p$ is an odd prime. We prove a mod $p$ formula analogous to a formula of Vistoli for the integral cohomology ring of $BPU(p)$. As an application, we give a simple topological proof of Vistoli's formula.