On $H^*(BPU_n; \mathbb{Z})$ and Weyl group invariants

TL;DR

This paper proves the surjectivity of the integral restriction homomorphism from the classifying space of the projective unitary group to Weyl group invariants and explores related ring isomorphisms and conditions for general compact Lie groups.

Contribution

It establishes the surjectivity of the integral restriction homomorphism for $PU_n$ and provides conditions for similar results in broader compact Lie groups.

Findings

The restriction homomorphism $ ho_{PU_n}$ is onto.

Several rings are isomorphic to $H^*(BT_{PU_n};bZ)^W$.

General conditions for the surjectivity of $ ho_G$ are given.

Abstract

For the projective unitary group $PU_n$ with a maximal torus $T_{PU_n}$ and Weyl group $W$, we show that the integral restriction homomorphism \[\rho_{PU_n} \colon H^*(BPU_n;\mathbb{Z})\rightarrow H^*(BT_{PU_n};\mathbb{Z})^W\] to the integral invariants of the Weyl group action is onto. We also present several rings naturally isomorphic to $H^*(BT_{PU_n};\mathbb{Z})^W$. In addition we give general sufficient conditions for the restriction homomorphism $\rho_G$ to be onto for a connected compact Lie group $G$.