Hopf algebra structure of the integral cohomology of PU_n for general n

Determine the Hopf algebra (including coproduct, counit, and antipode) structure on the graded integral cohomology H^*(PU_n; Z) of the projective unitary group PU_n for arbitrary positive integer n, so that computations via the Eilenberg–Moore spectral sequence for the universal principal PU_n-bundle can proceed in the integral setting.

Background

A standard approach to computing H^*(BG; R) for a topological group G uses the Eilenberg–Moore spectral sequence E_2 = Cotor_{H^*(G; R)}(R, R) ⇒ H^*(BG; R), which requires detailed knowledge of the Hopf algebra structure on H^*(G; R). For PU_n, the mod p Hopf algebra structure of H^*(PU_n; Z/p) is known (Baum–Browder), and the integral ring structure H^*(PU_n; Z) has been determined by Duan.

However, for integral cohomology H^*(PU_n; Z), the Hopf algebra structure itself is not available in general, preventing direct application of the Eilenberg–Moore spectral sequence to compute H^*(BPU_n; Z). The paper therefore adopts an alternative Serre spectral sequence approach via the fibration BU_n → BPU_n → K(Z,3).

The authors explicitly note this gap to explain why the usual Eilenberg–Moore strategy is impractical here, emphasizing the importance of determining the Hopf algebra structure of H^*(PU_n; Z) for general n.