Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Equivariant Seiberg-Witten-Floer cohomology

David Baraglia, Pedram Hekmati

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.

Equivariant Seiberg-Witten-Floer cohomology

Abstract

We develop an equivariant version of Seiberg-Witten-Floer cohomology for finite group actions on rational homology -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 -invariants which are indexed by the group cohomology of . These invariants satisfy a Froyshov-type inequality under equivariant cobordisms. Lastly we consider a variety of applications of these -invariants: concordance invariants of knots via branched covers, obstructions to extending group actions over bounding -manifolds, Nielsen realisation problems for -manifolds with boundary and obstructions to equivariant embeddings of -manifolds in -manifolds.

Paper Structure

This paper contains 39 sections, 46 theorems, 251 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Applications
  4. Non-extendable actions (Section \ref{['sec:app1']})
  5. Realisation problems (Section \ref{['sec:app2']})
  6. Equivariant embeddings of $3$-manifolds in $4$-manifolds (Section \ref{['sec:app3']})
  7. Comparison with other works
  8. Structure of the paper
  9. Seiberg--Witten--Floer spectra
  10. Seiberg--Witten trajectories
  11. Restriction to the global Coulomb slice
  12. Finite dimensional approximation
  13. The Conley index
  14. Equivariant Spanier--Whitehead category
  15. Seiberg--Witten--Floer cohomology
  16. ...and 24 more sections

Key Result

Theorem 1.1

There is a spectral sequence $E_r^{p,q}$ abutting to $HSW^*_G(Y , \mathfrak{s})$ whose second page is given by

Theorems & Definitions (122)

  • Theorem 1.1: Spectral sequence
  • Theorem 1.2: Localisation
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5: Equivariant Frø yshov inequality
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.8
  • proof
  • Example 1.9
  • ...and 112 more