The Kervaire-Milnor invariant in the stable classification of spin 4-manifolds
The Kervaire-Milnor invariant in the stable classification of spin 4-manifolds
We consider the role of the Kervaire-Milnor invariant in the classification of closed, connected, spin 4-manifolds up to stabilisation by connected sums with copies of $S^2 \times S^2$. This stable classification is detected by a spin bordism group over the classifying space $B\pi$ of the fundamental group. Part of the computation of this bordism group via an Atiyah-Hirzebruch spectral sequence is determined by a collection of codimension two Arf invariants. We show that these Arf invariants can be computed by the Kervaire-Milnor invariant evaluated on certain elements of $\pi_2$. In particular this yields a new stable classification of spin 4-manifolds with 2-dimensional fundamental groups.
math.GT, 57K40
Powell, Mark
4aaeb063-734c-4136-9da8-fb2ade23d744
Kasprowski, Daniel
44af11b9-4d22-49f2-a6a3-04009f45b075
Teichner, Peter
aee2a853-8909-43cc-a810-b0a8e10fc77b
Powell, Mark
4aaeb063-734c-4136-9da8-fb2ade23d744
Kasprowski, Daniel
44af11b9-4d22-49f2-a6a3-04009f45b075
Teichner, Peter
aee2a853-8909-43cc-a810-b0a8e10fc77b
[Unknown type: UNSPECIFIED]
Abstract
We consider the role of the Kervaire-Milnor invariant in the classification of closed, connected, spin 4-manifolds up to stabilisation by connected sums with copies of $S^2 \times S^2$. This stable classification is detected by a spin bordism group over the classifying space $B\pi$ of the fundamental group. Part of the computation of this bordism group via an Atiyah-Hirzebruch spectral sequence is determined by a collection of codimension two Arf invariants. We show that these Arf invariants can be computed by the Kervaire-Milnor invariant evaluated on certain elements of $\pi_2$. In particular this yields a new stable classification of spin 4-manifolds with 2-dimensional fundamental groups.
Text
2105.12153
- Author's Original
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Accepted/In Press date: 25 May 2021
Keywords:
math.GT, 57K40
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Local EPrints ID: 475972
URI: http://eprints.soton.ac.uk/id/eprint/475972
ISSN: 2331-8422
PURE UUID: 752bb0d4-9037-49e7-8a76-ad34d4634288
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Date deposited: 03 Apr 2023 16:37
Last modified: 17 Mar 2024 04:19
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Author:
Mark Powell
Author:
Daniel Kasprowski
Author:
Peter Teichner
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