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    Cup-length obstruction from a contractible cover

    P-QUAL-REVIEW-HATCHER-03

    Area
    Topology
    Status
    draft
    Contents

    Let \(X\) be a topological space with subspaces \(A_1,\dots,A_n\). Assume that every \(A_i\) is contractible and that \[ \bigcup_i A_i=X. \]

    1. Prove that if \(R\) is a ring and \(\alpha_1,\dots,\alpha_n\in H^*(X;R)\) all have positive degree, then \[ \alpha_1\smile\cdots\smile\alpha_n=0. \]

    2. Prove that \(\mathbb{RP}^n\) cannot be covered by \(n\) contractible subspaces.

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