[Paper Review] The equivalence of Heegaard Floer homology and embedded contact homology III: from hat to plus

Vincent Colin, Paolo Ghiggini+1 more|arXiv (Cornell University)|Aug 7, 2012

Geometric and Algebraic Topology36 references40 citations

TL;DR

This paper establishes a quasi-isomorphism between the Heegaard Floer homology group $HF^+(-M)$ and the embedded contact homology group $ECH(M)$ for any closed oriented 3-manifold $M$. Using an open book decomposition $(S, ho)$, the authors construct a chain map $\Phi^+$ from a Heegaard Floer chain complex to an embedded contact homology complex that commutes with the $U$-maps up to homotopy, proving the two invariants are isomorphic via algebraic topology arguments.

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.

Motivation & Objective

• To establish a quasi-isomorphism between the $HF^+$ and $ECH$ invariants of a closed oriented 3-manifold $M$.
• To extend the previously constructed quasi-isomorphism $\Phi$ for the hat versions ($\widehat{HF}$ and $\widehat{ECH}$) to the plus versions.
• To define a chain map $\Phi^+$ that commutes with the $U$-maps on both sides up to homotopy.
• To prove that $\Phi^+$ induces an isomorphism on homology, thereby establishing the equivalence of $HF^+$ and $ECH$.
• To reconcile the geometric construction of $\Phi^+$ with the algebraic structure of the $U$-action using filtered chain complexes.

Proposed method

• Construct a symplectic cobordism $X_+$ from $[0,1] \times \Sigma$ to $M$, extending a previous cobordism $W_+$ used in the hat version.
• Define the chain map $\Phi^+$ using a geometric cobordism $X_+$ that incorporates the mapping torus of the monodromy and a cylindrical end over $S_{1/2}$.
• Use an open book decomposition $(S, \rho)$ with $g \geq 2$ to define the Heegaard surface $\Sigma = S_0 \cup -S_{1/2}$ and associated chain complexes.
• Define a filtered chain complex $C(U)$ modeling $HF^+$ and a corresponding $C(U')$ for $ECH(M)$, with a filtration $\widehat{\mathcal{F}}$ on $C(U)$.
• Construct an algebraic map $\Phi_{\text{alg}}$ between these filtered complexes and show it is a quasi-isomorphism via spectral sequence arguments.
• Prove that $\Phi^+$ induces a map on homology that is isomorphic to the known quasi-isomorphism $\Phi_*$ on the hat versions, using commutative diagrams and homotopy data.

Experimental results

Research questions

• RQ1 Does there exist a chain map $\Phi^+$ from the Heegaard Floer complex $CF^+(-M)$ to the embedded contact homology complex $ECC(M)$ that commutes with the $U$-maps up to homotopy?
• RQ2 Can the isomorphism between $\widehat{HF}(-M)$ and $\widehat{ECH}(M)$ be extended to the $+$-versions via a geometrically defined chain map?
• RQ3 How can the $U$-action on $HF^+$ be algebraically reconciled with the $U$-action on $ECH$ through a geometric cobordism?
• RQ4 Is the induced map $\Phi^+$ on homology an isomorphism, thereby proving the equivalence of $HF^+$ and $ECH$?
• RQ5 Can the construction of $\Phi^+$ be carried out using twisted coefficients, as in prior work?

Key findings

• The chain map $\Phi^+$ is a quasi-isomorphism between $CF^+(-M)$ and $ECC(M)$, establishing the isomorphism $HF^+(-M) \cong ECH(M)$.
• The map $\Phi^+$ commutes with the $U$-maps on both sides up to a specified chain homotopy $K^+ = K + \Phi^+ \circ H$, where $H$ and $K$ are defined in earlier theorems.
• The algebraic map $\Phi_{\text{alg}}$ between filtered chain complexes is a quasi-isomorphism, as shown by spectral sequence arguments and commutativity with known isomorphisms.
• The construction of $\Phi^+$ requires the use of both $S_0$ and $S_{1/2}$ in the Heegaard surface $\Sigma = S_0 \cup -S_{1/2}$, unlike the hat version which only used $S_0$.
• The map $\Phi^+$ extends the previously constructed $\Phi$ for the hat versions, and the induced map on homology agrees with $\Phi_*$ on the subcomplexes defining $\widehat{HF}$ and $\widehat{ECH}$.
• The proof relies on the fact that the induced map on the $E^1$-page of the spectral sequence is an isomorphism, which implies the original map $\mathfrak{i}$ and hence $\Phi_{\text{alg}}$ are quasi-isomorphisms.