The equivalence of Heegaard Floer homology and embedded contact homology III: from hat to plus

Abstract

Given a closed oriented 3-manifold M, we establish an isomorphism between the Heegaard Floer homology group HF^+(-M) and the embedded contact homology group ECH(M). Starting from an open book decomposition (S,h) of M, we construct a chain map Φ^+ from a Heegaard Floer chain complex associated to (S,h) to an embedded contact homology chain complex for a contact form supported by (S,h). The chain map Φ^+ commutes up to homotopy with the U-maps defined on both sides and reduces to the quasi-isomorphism Φfrom "The equivalence of Heegaard Floer homology and embedded contact homology I, II" on subcomplexes defining the hat versions. Algebraic considerations then imply that the map Φ^+ is a quasi-isomorphism.

32 pages, 3 figures. Second version: Notation and references updated to match Part I and II

References in corpus (6)

• HF=HM I : Heegaard Floer homology and Seiberg--Witten Floer homology
• Contact structures, sutured Floer homology and TQFT
• Proof of the Arnold chord conjecture in three dimensions II
• HF=HM III: Holomorphic curves and the differential for the ech/Heegaard Floer correspondence
• HF=HM IV: The Seiberg-Witten Floer homology and ech correspondence
• HF=HM V: Seiberg-Witten-Floer homology and handle addition

Cited by in corpus (22)

• The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions II
• A Hierarchy of Local Symplectic Filling Obstructions for Contact 3-Manifolds
• A contact invariant in sutured monopole homology
• Graph cobordisms and Heegaard Floer homology
• Instanton Floer homology, sutures, and Heegaard diagrams
• Bordered Floer homology and incompressible surfaces
• Duality and mapping tori in Heegaard Floer homology
• On the equivalence of contact invariants in sutured Floer homology theories
• HF = ECH via open book decompositions: a summary
• Equivariant Seiberg-Witten-Floer cohomology
• An enhanced Euler characteristic of sutured instanton homology
• The Seiberg-Witten equations and the length spectrum of hyperbolic three-manifolds
• Noncommutative Hodge-to-de Rham spectral sequence and the Heegaard Floer homology of double covers
• Dimensionally-reduced sutured Floer homology as a string homology
• Cobordism maps in embedded contact homology
• On contact type hypersurfaces in 4-space
• On symplectic fillings of spinal open book decompositions II: Holomorphic curves and classification
• Applications of higher-dimensional Heegaard Floer homology to contact topology
• Seiberg-Witten Floer homotopy contact invariant
• Formal Contact Categories
• An exposition of the equivalence of Heegaard Floer homology and embedded contact homology
• Foliations, contact structures and their interactions in dimension three