Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Real Heegaard Floer Homology

Gary Guth, Ciprian Manolescu

TL;DR

This work constructs a real analogue of Heegaard Floer homology for 3-manifolds with an involution whose fixed set has codimension two, situating it as a special case of real Lagrangian Floer theory. It develops real invariants HFR^-, HFR^∞, HFR^+, and ŜHFR by counting R-invariant holomorphic strips in a symplectic setting on symmetric products, with careful attention to real Maslov indices, gradings, Spin^c structures, and Euler structures. The theory is proven invariant under the natural real Heegaard moves and almost complex structure changes, and is illustrated with branched double covers and explicit computations of Euler characteristics that connect to Miyazawa-type knot invariants. It also outlines potential connections to four-dimensional cobordisms, Khovanov-type theories, and standard Heegaard Floer groups, while highlighting rich structures in the real setting such as curved complexes and refined gradings. Overall, the paper provides a foundations and computational framework for real Heegaard Floer theory with broad implications for real gauge theory and knot theory via branched covers.

Abstract

We define an invariant of three-manifolds with an involution with non-empty fixed point set of codimension $2$; in particular, this applies to double branched covers over knots. Our construction gives the Heegaard Floer analogue of Li's real monopole Floer homology. It is a special case of a real version of Lagrangian Floer homology, which may be of independent interest to symplectic geometers. The Euler characteristic of the real Heegaard Floer homology is the analogue of Miyazawa's invariant, and can be computed combinatorially for all knots.

Real Heegaard Floer Homology

TL;DR

This work constructs a real analogue of Heegaard Floer homology for 3-manifolds with an involution whose fixed set has codimension two, situating it as a special case of real Lagrangian Floer theory. It develops real invariants HFR^-, HFR^∞, HFR^+, and ŜHFR by counting R-invariant holomorphic strips in a symplectic setting on symmetric products, with careful attention to real Maslov indices, gradings, Spin^c structures, and Euler structures. The theory is proven invariant under the natural real Heegaard moves and almost complex structure changes, and is illustrated with branched double covers and explicit computations of Euler characteristics that connect to Miyazawa-type knot invariants. It also outlines potential connections to four-dimensional cobordisms, Khovanov-type theories, and standard Heegaard Floer groups, while highlighting rich structures in the real setting such as curved complexes and refined gradings. Overall, the paper provides a foundations and computational framework for real Heegaard Floer theory with broad implications for real gauge theory and knot theory via branched covers.

Abstract

We define an invariant of three-manifolds with an involution with non-empty fixed point set of codimension ; in particular, this applies to double branched covers over knots. Our construction gives the Heegaard Floer analogue of Li's real monopole Floer homology. It is a special case of a real version of Lagrangian Floer homology, which may be of independent interest to symplectic geometers. The Euler characteristic of the real Heegaard Floer homology is the analogue of Miyazawa's invariant, and can be computed combinatorially for all knots.

Paper Structure

This paper contains 35 sections, 38 theorems, 183 equations, 16 figures, 1 table.

Table of Contents

  1. Introduction
  2. Real Lagrangian Floer homology
  3. The set-up
  4. Transversality
  5. The real Maslov index
  6. The real Lagrangian Floer complex
  7. Gradings
  8. Real Heegaard diagrams
  9. Real Heegaard Diagrams
  10. Real Heegaard Moves
  11. Multi-pointed real Heegaard diagrams
  12. Real Invariant Domains
  13. Real $\mathrm{Spin}^{c}$-structures
  14. Real Euler Structures
  15. Real Heegaard diagrams and real $\mathrm{Spin}^{c}$-structures
  16. ...and 20 more sections

Key Result

Theorem 1

The isomorphism classes of the $\mathbb{F}[U]\otimes \Lambda^*(H_1(Y,\mathbb{Z})^{-\tau_*}/\mathrm{Tors})$-modules $\mathit{HFR}^\circ(Y, \tau, w)$ are topological invariants of the underlying pointed real 3-manifold $(Y, \tau, w)$.

Figures (16)

  • Figure 2.1: A bigon domain from a path from $\Lambda_0 = \Gamma(-I)$ to $\Lambda_1=\Gamma(I)$.
  • Figure 3.1: (a) A band $B$ and its dual band $B^*$ attached to the surface $F_0$; (b) The resulting thickened surface.
  • Figure 3.2: Left: a $\mathbb{Z}/2$-surface with orientable quotient. The involution is given by reflection. Right: A $\mathbb{Z}/2$-surface with nonorientable quotient; the involution is reflection on the complement of the handle $h$ and the antipodal map on $h$.
  • Figure 3.3: Three real Heegaard diagrams for $(S^3, \tau)$. The fixed sets of the various involutions are shown in green.
  • Figure 3.4: Top: A free stabilization. Bottom: A fixed point stabilization.
  • ...and 11 more figures

Theorems & Definitions (118)

  • Theorem 1
  • Theorem 2
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Lemma 2.5
  • proof
  • ...and 108 more