Homology manifolds and euclidean bundles

We construct a Poincar\'e complex whose periodic total surgery obstruction vanishes but whose Spivak normal fibration does not admit a reduction to a stable euclidean bundle. This contradicts the conjunction of two claims in the literature: Namely, on the one hand that a Poincar\'e complex...

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Bibliographic Details
Main Authors Hebestreit, Fabian, Land, Markus, Weiss, Michael, Winges, Christoph
Format Journal Article
LanguageEnglish
Published 20.06.2024
Subjects
Mathematics - Algebraic Topology
Mathematics - Geometric Topology
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Summary:We construct a Poincar\'e complex whose periodic total surgery obstruction vanishes but whose Spivak normal fibration does not admit a reduction to a stable euclidean bundle. This contradicts the conjunction of two claims in the literature: Namely, on the one hand that a Poincar\'e complex with vanishing periodic total surgery obstruction is homotopy equivalent to a homology manifold, which appears in work of Bryant--Ferry--Mio--Weinberger, and on the other that the Spivak normal fibration of a homology manifold always admits a reduction to a stable euclidean bundle, which appears in work of Ferry--Pedersen.
DOI:10.48550/arxiv.2406.14677