The Homotopy Type of a Poincar\'e Duality Complex after Looping

We answer a weaker version of the classification problem for the homotopy types of $(n-2)$-connected closed orientable $(2n-1)$-manifolds. Let $n\geq 6$ be an even integer, and $X$ be a $(n-2)$-connected finite orientable Poincar\'e $(2n-1)$-complex such that $H^{n-1}(X;\mathbb{Q})=0$ and $H^{n...

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Bibliographic Details
Main Authors Beben, Piotr, Wu, Jie
Format Journal Article
LanguageEnglish
Published 08.02.2011
Subjects
Mathematics - Algebraic Topology
Mathematics - Geometric Topology
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Summary:We answer a weaker version of the classification problem for the homotopy types of $(n-2)$-connected closed orientable $(2n-1)$-manifolds. Let $n\geq 6$ be an even integer, and $X$ be a $(n-2)$-connected finite orientable Poincar\'e $(2n-1)$-complex such that $H^{n-1}(X;\mathbb{Q})=0$ and $H^{n-1}(X;\mathbb{Z}_2)=0$. Then its loop space homotopy type is uniquely determined by the action of higher Bockstein operations on $H^{n-1}(X;\mathbb{Z}_p)$ for each odd prime $p$. A stronger result is obtained when localized at odd primes.
DOI:10.48550/arxiv.1102.1516