Algebraic criteria for stable diffeomorphism of spin 4-manifolds
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arXiv
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| Natura: | Preprint |
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2020
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| author | Kasprowski, Daniel Powell, Mark Teichner, Peter |
| author_facet | Kasprowski, Daniel Powell, Mark Teichner, Peter |
| contents | We study closed, connected, spin 4-manifolds up to stabilisation by connected sums with copies of $S^2 \times S^2$. For a fixed fundamental group, there are primary, secondary and tertiary obstructions, which together with the signature lead to a complete stable classification. The primary obstruction exactly detects $\mathbb{CP}^2$-stable diffeomorphism and was previously related to algebraic invariants by Kreck and the authors.
In this article we formulate conjectural relationships of the secondary and tertiary obstructions with algebraic invariants: the secondary obstruction should be determined by the (stable) equivariant intersection form and the tertiary obstruction via a $τ$-invariant recording intersection data between 2-spheres, with trivial algebraic self-intersection, and their Whitney discs.
We prove our conjectures for the following classes of fundamental groups: groups of cohomological dimension at most 3, right-angled Artin groups, abelian groups, and finite groups with quaternion or abelian 2-Sylow subgroups. We apply our theory to give a complete algebraic stable classification of spin $4$-manifolds with fundamental group $\mathbb{Z} \times \mathbb{Z}/2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2006_06127 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Algebraic criteria for stable diffeomorphism of spin 4-manifolds Kasprowski, Daniel Powell, Mark Teichner, Peter Geometric Topology Algebraic Topology 57K40 We study closed, connected, spin 4-manifolds up to stabilisation by connected sums with copies of $S^2 \times S^2$. For a fixed fundamental group, there are primary, secondary and tertiary obstructions, which together with the signature lead to a complete stable classification. The primary obstruction exactly detects $\mathbb{CP}^2$-stable diffeomorphism and was previously related to algebraic invariants by Kreck and the authors. In this article we formulate conjectural relationships of the secondary and tertiary obstructions with algebraic invariants: the secondary obstruction should be determined by the (stable) equivariant intersection form and the tertiary obstruction via a $τ$-invariant recording intersection data between 2-spheres, with trivial algebraic self-intersection, and their Whitney discs. We prove our conjectures for the following classes of fundamental groups: groups of cohomological dimension at most 3, right-angled Artin groups, abelian groups, and finite groups with quaternion or abelian 2-Sylow subgroups. We apply our theory to give a complete algebraic stable classification of spin $4$-manifolds with fundamental group $\mathbb{Z} \times \mathbb{Z}/2$. |
| title | Algebraic criteria for stable diffeomorphism of spin 4-manifolds |
| topic | Geometric Topology Algebraic Topology 57K40 |
| url | https://arxiv.org/abs/2006.06127 |