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, typically denoted by M, up to stabilisation by connected sums with copies of S2 x S2. This stable classification is detected by a spin bordism group over the classifying space Bπ 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 π2 (M). In particular this yields a new stable classification of spin 4-manifolds with 2-dimensional fundamental groups, namely those for which Bπ admits a finite 2-dimensional CW-complex model.
57K40, math.GT, 4-manifolds, stable diffeomorphism, Kervaire–Milnor invariant
417-436
Kasprowski, Daniel
44af11b9-4d22-49f2-a6a3-04009f45b075
Powell, Mark
1361d6ba-3a6c-4ce2-9dc4-f8e1bd678239
Teichner, Peter
aee2a853-8909-43cc-a810-b0a8e10fc77b
9 May 2025
Kasprowski, Daniel
44af11b9-4d22-49f2-a6a3-04009f45b075
Powell, Mark
1361d6ba-3a6c-4ce2-9dc4-f8e1bd678239
Teichner, Peter
aee2a853-8909-43cc-a810-b0a8e10fc77b
Kasprowski, Daniel, Powell, Mark and Teichner, Peter
(2025)
The Kervaire-Milnor invariant in the stable classification of spin 4-manifolds.
Tunisian Journal of Mathematics, 7 (2), .
(doi:10.2140/tunis.2025.7.417).
Abstract
We consider the role of the Kervaire–Milnor invariant in the classification of closed, connected, spin 4-manifolds, typically denoted by M, up to stabilisation by connected sums with copies of S2 x S2. This stable classification is detected by a spin bordism group over the classifying space Bπ 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 π2 (M). In particular this yields a new stable classification of spin 4-manifolds with 2-dimensional fundamental groups, namely those for which Bπ admits a finite 2-dimensional CW-complex model.
Text
2105.12153
- Author's Original
Text
2105.12153v3
- Accepted Manuscript
More information
Accepted/In Press date: 31 October 2024
Published date: 9 May 2025
Keywords:
57K40, math.GT, 4-manifolds, stable diffeomorphism, Kervaire–Milnor invariant
Identifiers
Local EPrints ID: 475972
URI: http://eprints.soton.ac.uk/id/eprint/475972
ISSN: 2576-7658
PURE UUID: 752bb0d4-9037-49e7-8a76-ad34d4634288
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Date deposited: 03 Apr 2023 16:37
Last modified: 30 Aug 2025 02:14
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Contributors
Author:
Daniel Kasprowski
Author:
Mark Powell
Author:
Peter Teichner
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