The Kervaire-Milnor invariant in the stable classification of spin 4-manifolds
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2021
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| _version_ | 1866909607923810304 |
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| author | Kasprowski, Daniel Powell, Mark Teichner, Peter |
| author_facet | Kasprowski, Daniel Powell, Mark Teichner, Peter |
| contents | 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 $S^2 \times S^2$. 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 $π_2(M)$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_12153 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The Kervaire-Milnor invariant in the stable classification of spin 4-manifolds Kasprowski, Daniel Powell, Mark Teichner, Peter Geometric Topology 57K40 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 $S^2 \times S^2$. 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 $π_2(M)$. 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. |
| title | The Kervaire-Milnor invariant in the stable classification of spin 4-manifolds |
| topic | Geometric Topology 57K40 |
| url | https://arxiv.org/abs/2105.12153 |