Equivariant Seiberg-Witten-Floer cohomology
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866910383398191104 |
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| author | Baraglia, David Hekmati, Pedram |
| author_facet | Baraglia, David Hekmati, Pedram |
| contents | We develop an equivariant version of Seiberg-Witten-Floer cohomology for finite group actions on rational homology $3$-spheres. Our construction is based on an equivariant version of the Seiberg-Witten-Floer stable homotopy type, as constructed by Manolescu. We use these equivariant cohomology groups to define a series of $d$-invariants $d_{G,c}(Y,\mathfrak{s})$ which are indexed by the group cohomology of $G$. These invariants satisfy a Froyshov-type inequality under equivariant cobordisms. Lastly we consider a variety of applications of these $d$-invariants: concordance invariants of knots via branched covers, obstructions to extending group actions over bounding $4$-manifolds, Nielsen realisation problems for $4$-manifolds with boundary and obstructions to equivariant embeddings of $3$-manifolds in $4$-manifolds. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2108_06855 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Equivariant Seiberg-Witten-Floer cohomology Baraglia, David Hekmati, Pedram Geometric Topology Differential Geometry 57M27, 57M60, 57R58 We develop an equivariant version of Seiberg-Witten-Floer cohomology for finite group actions on rational homology $3$-spheres. Our construction is based on an equivariant version of the Seiberg-Witten-Floer stable homotopy type, as constructed by Manolescu. We use these equivariant cohomology groups to define a series of $d$-invariants $d_{G,c}(Y,\mathfrak{s})$ which are indexed by the group cohomology of $G$. These invariants satisfy a Froyshov-type inequality under equivariant cobordisms. Lastly we consider a variety of applications of these $d$-invariants: concordance invariants of knots via branched covers, obstructions to extending group actions over bounding $4$-manifolds, Nielsen realisation problems for $4$-manifolds with boundary and obstructions to equivariant embeddings of $3$-manifolds in $4$-manifolds. |
| title | Equivariant Seiberg-Witten-Floer cohomology |
| topic | Geometric Topology Differential Geometry 57M27, 57M60, 57R58 |
| url | https://arxiv.org/abs/2108.06855 |