Equivariant Seiberg-Witten theory
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911898973241344 |
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| author | Baraglia, David |
| author_facet | Baraglia, David |
| contents | We introduce and study equivariant Seiberg-Witten invariants for $4$-manifolds equipped with a smooth action of a finite group $G$. Our invariants come in two types: cohomological, valued in the group cohomology of $G$ and $K$-theoretic, valued in the representation ring of $G$. We establish basic properties of the invariants such as wall-crossing and vanishing of the invariants for $G$-invariant positive scalar curvature metrics. We establish a relation between the equivariant Seiberg-Witten invariants and families Seiberg-Witten invariants. Sufficient conditions are found under which equivariant transversality can be achieved leading to smooth moduli spaces on which $G$ acts. In the zero-dimensional case this yields a further invariant of the $G$-action valued in a refinement of the Burnside ring of $G$. We prove localisation formulas in cohomology and $K$-theory, relating the equivariant Seiberg-Witten invariants to moduli spaces of $G$-invariant solutions. We give an explicit formula for the invariants for holomorphic group actions on Kähler surfaces. We also prove a gluing formula for the invariants of equivariant connected sums. Various applications and consequences of the theory are considered. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_00642 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Equivariant Seiberg-Witten theory Baraglia, David Differential Geometry Geometric Topology We introduce and study equivariant Seiberg-Witten invariants for $4$-manifolds equipped with a smooth action of a finite group $G$. Our invariants come in two types: cohomological, valued in the group cohomology of $G$ and $K$-theoretic, valued in the representation ring of $G$. We establish basic properties of the invariants such as wall-crossing and vanishing of the invariants for $G$-invariant positive scalar curvature metrics. We establish a relation between the equivariant Seiberg-Witten invariants and families Seiberg-Witten invariants. Sufficient conditions are found under which equivariant transversality can be achieved leading to smooth moduli spaces on which $G$ acts. In the zero-dimensional case this yields a further invariant of the $G$-action valued in a refinement of the Burnside ring of $G$. We prove localisation formulas in cohomology and $K$-theory, relating the equivariant Seiberg-Witten invariants to moduli spaces of $G$-invariant solutions. We give an explicit formula for the invariants for holomorphic group actions on Kähler surfaces. We also prove a gluing formula for the invariants of equivariant connected sums. Various applications and consequences of the theory are considered. |
| title | Equivariant Seiberg-Witten theory |
| topic | Differential Geometry Geometric Topology |
| url | https://arxiv.org/abs/2406.00642 |