On 𝐻*(𝐵𝑃𝑈𝑛;ℤ) and Weyl group invariants

For the projective unitary group P U n PU_n with a maximal torus T P U n TPU_n and Weyl group W W , we show that the integral restriction homomorphism \[ ρ P U n : H ∗ ( B P U n ; Z ) → H ∗ ( B T P U n ; Z ) W ρ PU_n H^*(BPU_n;Z)→ H^*(BTPU_n;Z)^W \] to the integral invariants of the Weyl group action is onto. We also present several rings naturally isomorphic to H ∗ ( B T P U n ; Z ) W H^*(BTPU_n;Z)^W . In addition we give general sufficient conditions for the restriction homomorphism ρ G ρ _G to be onto for a connected compact Lie group G G .