On $H^*(BPU_n; \mathbb{Z})$ and Weyl group invariants
Diarmuid Crowley, Xing Gu
TL;DR
This work addresses the problem of understanding integral Weyl-invariant cohomology in classifying spaces by proving that the restriction map $ρ_{PU_n}: H^*(BPU_n) o H^*(BT_{PU_n})^W$ is onto for all $n$. The authors develop a general framework using weak $V$-$CW$-models and homotopy orbit fibrations to formulate and verify sufficient conditions for surjectivity, then construct a $ ext{ψ}$-adapted weak $W$-$CW$-model for $BT_{PU_n}$ via Milgram’s model and a $W$-equivariant map from $P^{ imes n}$ to $B_{ ext{M}}T^{n-1}$. They reveal a rich, equivalent description of Weyl invariants through multiple algebraic/topological avatars: the torsion-free quotient $FH^*(BPU_n)$, the kernel of the derivation $ abla_n$ on $H^*(BU_n)$, the primitive subring $PH^*(BU_n)$, and the Leray–Serre $d^{0,*}_3$-kernel, establishing canonical isomorphisms among these descriptions. These results connect integral cohomology calculations for $BPU_n$ to the structure of $BU_n$ and its torus invariants, with implications for the topological period–index problem and potential canonical liftings of Weyl-invariant classes. Overall, the paper advances a coherent, multi-faceted picture of Weyl invariants in the integral setting and provides practical criteria to verify surjectivity of restriction maps in broader classes of compact Lie groups.
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 \[ρ_{PU_n} \colon H^*(BPU_n;\mathbb{Z})\rightarrow H^*(BT_{PU_n};\mathbb{Z})^W\] 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 $ρ_G$ to be onto for a connected compact Lie group $G$.