Torsion in the integral cohomology of $BPU_{n}$

Question

I would like to prove that the integral cohomology of $BPU_{n}$ the classifying space of the projective unitary group of order $n$ has $n-$primary torsion.

We have a fiber sequence of the form $BSU_{n}\rightarrow BPU_{n} \rightarrow K(\mathbb{Z}/n,2)$. Then we can consider the Serre spectral sequence associated to this fibration. Let's suppose that the smallest prime dividing $n$ is a very large prime $p$. For general reasons, we know that the integral cohomology of $H^{i}(K(\mathbb{Z}/n,2))$ is $\mathbb{Z}$, 0, 0, $\mathbb{Z}/n$, 0, $\mathbb{Z}/n$, 0, $\mathbb{Z}/n$, ... where it repeats this way in a range $2p+\delta$. We also know that $H^{*}(BSU_{n})\cong \mathbb{Z}[c_{2},\dots,c_{n}]$ where deg$(c_{i})=2i$. Therefore, third page of the spectral sequences will have all its columns (but the zeroth column) conformed by zeros and cyclic groups of order $n$:

How can I answer my question using this setup?

Thanks.

Answer 1

You may want to have a look at this paper:

• X. Gu. On the cohomology of classifying spaces of projective unitary groups. arXiv:1612.00506, (link to arXiv)

The spectral sequence involving $BSU_n$ appears in the discussion in Section 5. The paper contains a lot of spectral sequence calculations, computations of differentials and determines cohomology (including ring structure) up to degree 10. Results like "the only torsion is $n$-torsion" and "there exists some nontrivial $n$-torsion" follow from these computations. Further references to other computations can also be found in the paper.