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The equivalence between two real Seiberg-Witten homologies

Yonghan Xiao

TL;DR

This work establishes a rigorous isomorphism between the real monopole Floer homology ${\mathrm{HMR}}$ and the real Seiberg–Witten Floer homotopy type ${\mathrm{SWF}}_{\mathbb{Z}_2}$ for real rational homology spheres with compatible real spin^c structures, unifying two parallel real-structure approaches to Seiberg–Witten invariants. The proof adapts the finite-dimensional Morse framework to the real setting by working in the global Coulomb slice, developing real Hessians, and constructing a perturbation theory that respects the ${\mathbb{Z}}_2$-action, then relating finite- and infinite-dimensional Morse theories via a sequence of intermediate complexes. It also yields corollaries including Frøyshov-type invariants and Smith-type inequalities, and identifies the $v$-action on both sides, providing real analogues of classical localization phenomena. The results solidify the compatibility between gauge-theoretic invariants on real three-manifolds and their homotopy-type counterparts, with implications for real L-spaces and branched-cover knot theory. Overall, the paper extends the Hitchin–Friedman–Manolescu program to the real setting, offering tools for comparing real Floer-type invariants across different formalisms and enabling new constraints in real three-manifold topology.

Abstract

We show that for a real rational homology sphere $Y$ equipped with a real $\mathrm{spin^c}$ structure $\s$, the real monopole Floer homology defined by Li and the real Seiberg-Witten Floer homology defined by Konno, Miyazawa and Taniguchi are isomorphic. As corollaries, we identify some Frøyshov-type invariants and prove two Smith-type inequalities.

The equivalence between two real Seiberg-Witten homologies

TL;DR

This work establishes a rigorous isomorphism between the real monopole Floer homology and the real Seiberg–Witten Floer homotopy type for real rational homology spheres with compatible real spin^c structures, unifying two parallel real-structure approaches to Seiberg–Witten invariants. The proof adapts the finite-dimensional Morse framework to the real setting by working in the global Coulomb slice, developing real Hessians, and constructing a perturbation theory that respects the -action, then relating finite- and infinite-dimensional Morse theories via a sequence of intermediate complexes. It also yields corollaries including Frøyshov-type invariants and Smith-type inequalities, and identifies the -action on both sides, providing real analogues of classical localization phenomena. The results solidify the compatibility between gauge-theoretic invariants on real three-manifolds and their homotopy-type counterparts, with implications for real L-spaces and branched-cover knot theory. Overall, the paper extends the Hitchin–Friedman–Manolescu program to the real setting, offering tools for comparing real Floer-type invariants across different formalisms and enabling new constraints in real three-manifold topology.

Abstract

We show that for a real rational homology sphere equipped with a real structure , the real monopole Floer homology defined by Li and the real Seiberg-Witten Floer homology defined by Konno, Miyazawa and Taniguchi are isomorphic. As corollaries, we identify some Frøyshov-type invariants and prove two Smith-type inequalities.

Paper Structure

This paper contains 44 sections, 91 theorems, 208 equations.

Table of Contents

  1. Introduction
  2. Convention
  3. Organization
  4. Review
  5. Morse homology and Morse quasi-gradient flow
  6. Real Seiberg-Witten homotopy type
  7. Configuration space and Coulomb slices
  8. The Seiberg-Witten equation
  9. Finite dimensional approximation
  10. The Conley index and the real Seiberg-Witten Floer spectrum
  11. Real monopole Floer homology
  12. Seiberg Witten equations on the blow-up
  13. Perturbations
  14. Real monopole Floer homology
  15. Gradings
  16. ...and 29 more sections

Key Result

Theorem 1.1

(Konno2024) Let $Y$ be a rational homology three sphere with a real structure $\iota$ and a compatible real $\mathrm{spin^c}$ structure $\mathfrak{s}$. Then we have an isomorphism of relatively graded $H_{{\mathbb{Z}}_2}^*(S^0;{\mathbb{F}})\cong {\mathbb{F}}[v]$-modules Here, $\widecheck{\mathit{HMR}}_{*}(Y,\iota,\mathfrak{s})$ is the "to" version of real monopole Floer homology defined in li2022

Theorems & Definitions (139)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Conjecture 1.7
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • ...and 129 more