Equivariant Seiberg-Witten-Floer cohomology

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 $d_{G,c}(Y,\mathfrak{s})$ 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.

58 pages, minor corrections

Cited by in corpus (4)

• Knot concordance invariants from Seiberg-Witten theory and slice genus bounds in 4-manifolds
• Brieskorn spheres, cyclic group actions and the Milnor conjecture
• Involutions, knots, and Floer K-theory
• Cables of the figure-eight knot via real Frøyshov invariants