The \(p\)-primary subgroups of the cohomology of \(BPU_n\) in dimensions less than \(2p+5\). (English) Zbl 1506.55013
Let \(U_{n}\) be the unitary group of \(n\times n\) matrices and let \(PU_{n}\) be the quotient of \(U_{n}\) by the natural action of the unit circle \(S^{1}\). The purpose of this paper is to compute the integral cohomology, \(H^{*}(BPU_{n};\mathbb{Z})\), up to a certain range. The result is the following Theorem. Let \(p\) be an odd prime number and \(n=p^{r}m\) where \(m\) is a positive integer and \(p\) does not divide \(m\). Then, the \(p\)-primary subgroup, \(pH^{s}(BPU_{n};\mathbb{Z})\), of \(H^{s}(BPU_{n};\mathbb{Z})\) in dimension less than \(2p+5\) is as follows: for \(r>0\),
\[
pH^{s}(BPU_{n};\mathbb{Z})= \begin{cases} \mathbb{Z}/p^{r}, s=3;\\
\mathbb{Z}/p, s=2p+2\\
o, s<2p+5, s\not=3,2p+2. \end{cases}
\]
and \(pH^{s}(BPU_{n};\mathbb{Z})=0\) for all \(s\geq 0\) and \(r=0\).
The main tool to achieve these computations is the Serre spectral sequence associated to the fibrations \(BU_{n}\to BPU_{n}\to K(\mathbb{Z},3)\) and \(BT^{n}\to BPT^{n}\to K(\mathbb{Z},3)\), where \(T^{n}\) is the maximal torus in \(U_{n}\).
The main tool to achieve these computations is the Serre spectral sequence associated to the fibrations \(BU_{n}\to BPU_{n}\to K(\mathbb{Z},3)\) and \(BT^{n}\to BPT^{n}\to K(\mathbb{Z},3)\), where \(T^{n}\) is the maximal torus in \(U_{n}\).
Reviewer: Daniel Juan Pineda (Michoacán)
MSC:
| 55T10 | Serre spectral sequences |
| 55R35 | Classifying spaces of groups and \(H\)-spaces in algebraic topology |
| 55R40 | Homology of classifying spaces and characteristic classes in algebraic topology |
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