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On Steenrod 𝕃-homology, generalized manifolds, and surgery

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

Published online by Cambridge University Press:  20 March 2020

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 & Institute of Mathematics, Physics and Mechanics, 1000Ljubljana, Slovenia ([email protected])
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Abstract

The aim of this paper is to show the importance of the Steenrod construction of homology theories for the disassembly process in surgery on a generalized n-manifold Xn, in order to produce an element of generalized homology theory, which is basic for calculations. In particular, we show how to construct an element of the nth Steenrod homology group $H^{st}_{n} (X^{n}, \mathbb {L}^+)$, where 𝕃+ is the connected covering spectrum of the periodic surgery spectrum 𝕃, avoiding the use of the geometric splitting procedure, the use of which is standard in surgery on topological manifolds.

Keywords

Poincaré duality complexgeneralized manifoldSteenrod 𝕃-homologyperiodic surgery spectrum 𝕃fundamental complex𝕃-homology classQuinn index

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 63 , Issue 2 , May 2020 , pp. 579 - 607
Copyright
Copyright © Edinburgh Mathematical Society 2020

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Footnotes

Dedicated to the memory of Professor Andrew Ranicki (1948–2018)

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