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I am reading Adem's paper Periodic Complexes and Group actions. But I can't give an argument about a statement on spectral sequences.

Suppose you have an orientable fibration of CW-complexes like follows: $P_{k+n-1}S^{n-1}\to E\to X$ Let $\alpha \in H^{n}(X,\mathbb{Z})$ be the transgression of a generator of $H^{n-1}(P_{n+k-1}S^{n-1};\mathbb{Z})\simeq \mathbb{Z}$

Then we form a Serre spectral sequence with coefficients $\pi_{n+k}(S^{n-1})$. $E_{2}^{p,q}=H^{p}(X;H^{q}(P_{n+k-1}S^{n-1};\pi_{n+k}(S^{n-1}))$

Choose a $k$-invariant $g_{k}: P_{k+n-1}S^{n-1}\to K(\pi_{n+k}(S^{n-1}),n+k+1)$ which corresponds an cohomology class $\gamma_{k}\in H^{n+k+1}(P_{n+k-1}S^{n-1};\pi_{n+k}(S^{n-1})$

When $n$ is large enough, why $\alpha \cup d_{k+3}\gamma_{k} =0$?

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  • $\begingroup$ What does the notation $P_{k+n−1} S^{n−1}$ mean? $\endgroup$
    – Dan Ramras
    Commented May 22, 2017 at 20:14
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    $\begingroup$ it means Postnikov sections of $S^{n-1}$ $\endgroup$
    – Zhipeng Duan
    Commented May 22, 2017 at 22:29

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