On the Cohomology of the Classifying Spaces of Projective Unitary Groups

Xing Gu

doi:10.1142/S1793525320500211

paper

On the Cohomology of the Classifying Spaces of Projective Unitary Groups

arXiv:1612.00506 · doi:10.1142/S1793525320500211

Abstract

Let $B P U_{n}$ be the classifying space of $P U_{n}$ , the projective unitary group of order $n$ , for $n > 1$ . We use the Serre spectral sequence associated to a fiber sequence $B U_{n} \to B P U_{n} \to K (Z, 3)$ to determine the ring structure of $H^{*} (B P U_{n}; Z)$ up to degree $10$ , as well as a family of distinguished elements of $H^{2 p + 2} (B P U_{n}; Z)$ , for each prime divisor $p$ of $n$ . We also study the primitive elements of $H^{*} (B U_{n}; Z)$ as a comodule over $H^{*} (K (Z, 2); Z)$ , where the comodule structure is given by an action of $K (Z, 2) ≃ B S^{1}$ on $B U_{n}$ corresponding to the action of taking the tensor product of a complex line bundle and an $n$ dimensional complex vector bundle.

43 pages, 3 figures. Published version plus an appendix. Accepted by the Journal of Topology and Analysis

On the Cohomology of the Classifying Spaces of Projective Unitary Groups

Abstract

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