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Equivariant Seiberg–Witten–Floer cohomology

David Baraglia and Pedram Hekmati

Algebraic & Geometric Topology 24 (2024) 493–554

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 dG,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.

Seiberg–Witten, Floer homology, Conley index, equivariant cohomology
Mathematical Subject Classification
Primary: 57K31
Secondary: 57K10, 57K41
Received: 13 January 2022
Revised: 27 July 2022
Accepted: 25 August 2022
Published: 18 March 2024
David Baraglia
School of Mathematical Sciences
The University of Adelaide
Adelaide, SA
Pedram Hekmati
Department of Mathematics
The University of Auckland
New Zealand

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