Equivariant Seiberg–Witten–Floer cohomology

Baraglia, David; Hekmati, Pedram

doi:10.2140/agt.2024.24.493

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Equivariant Seiberg–Witten–Floer cohomology

David Baraglia and Pedram Hekmati

Algebraic & Geometric Topology 24 (2024) 493–554

DOI: 10.2140/agt.2024.24.493

Abstract

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 $d_{G, c} (Y, 𝔰)$ which are indexed by the group cohomology of $G$ . These invariants satisfy a Frøyshov-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.

Keywords

Seiberg–Witten, Floer homology, Conley index, equivariant cohomology

Mathematical Subject Classification

Primary: 57K31

Secondary: 57K10, 57K41

References

Publication

Received: 13 January 2022

Revised: 27 July 2022

Accepted: 25 August 2022

Published: 18 March 2024

Authors

David Baraglia
School of Mathematical Sciences The University of Adelaide Adelaide, SA Australia

Pedram Hekmati
Department of Mathematics The University of Auckland Auckland New Zealand

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Volume 24, issue 1 (2024)