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Generalized manifolds, normal invariants, and 𝕃-homology

Part of: Generalized manifolds Homology and cohomology theories Differential topology Classical Topics in Algebraic Topology Categories and geometry Fiber spaces and bundles

Published online by Cambridge University Press:  16 June 2021

Friedrich Hegenbarth  and
Dušan Repovš
Friedrich Hegenbarth
Affiliation:
Dipartimento di Matematica ‘Federigo Enriques’, Università degli studi di Milano, 20133Milano, Italy ([email protected])
Dušan Repovš
Affiliation:
Faculty of Education and Faculty of Mathematics and Physics, University of Ljubljana, 1000Ljubljana, Slovenia ([email protected])
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Abstract

Let $X^{n}$ be an oriented closed generalized $n$-manifold, $n\ge 5$. In our recent paper (Proc. Edinb. Math. Soc. (2) 63 (2020), no. 2, 597–607), we have constructed a map $t:\mathcal {N}(X^{n}) \to H^{st}_{n} ( X^{n}; \mathbb{L}^{+})$ which extends the normal invariant map for the case when $X^{n}$ is a topological $n$-manifold. Here, $\mathcal {N}(X^{n})$ denotes the set of all normal bordism classes of degree one normal maps $(f,\,b): M^{n} \to X^{n},$ and $H^{st}_{*} ( X^{n}; \mathbb{E})$ denotes the Steenrod homology of the spectrum $\mathbb{E}$. An important non-trivial question arose whether the map $t$ is bijective (note that this holds in the case when $X^{n}$ is a topological $n$-manifold). It is the purpose of this paper to prove that the answer to this question is affirmative.

Keywords

Generalized manifoldSteenrod $\mathbb{L}$-homologyPoincaré duality complexnormal invariant of degree one mapperiodic surgery spectrum $\mathbb{L}$fundamental complexSpivak fibrationPontryagin–Thom constructionSpanier–Whitehead dualityabsolute neighbourhood retract

MSC classification

Primary: 55N07: Steenrod-Sitnikov homologies 55R20: Spectral sequences and homology of fiber spaces 57P10: Poincaré duality spaces 57R67: Surgery obstructions, Wall groups
Secondary: 18F15: Abstract manifolds and fiber bundles 55M05: Duality 55N20: Generalized (extraordinary) homology and cohomology theories 57P05: Local properties of generalized manifolds 57P99: None of the above, but in this section 57R65: Surgery and handlebodies
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 64 , Issue 3 , August 2021 , pp. 574 - 589
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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Footnotes

Dedicated to the memory of Professor Erik Kjær Pedersen (1946–2020)

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