[Paper Review] Heegaard Floer homology and Morse surgery

Eaman Eftekhary|ArXiv.org|Mar 7, 2006

Geometric and Algebraic Topology5 references5 citations

TL;DR

This paper establishes a surgery formula for the filtration on the Heegaard Floer homology complex induced by the core of a surgery torus in rational surgeries on null-homologous knots in homology spheres. Using a mapping cone construction involving a filtered chain complex and a twisted chain map, it computes the knot Floer homology of the surgered manifold, proving that for a knot $K$ in $S^3$ of genus $g(K)$, the homology $\widehat{\text{HFK}}(K_r, \underline{\mathfrak{t}})$ is non-vanishing precisely at $\underline{\mathfrak{t}} = -qg(K)$ and $\underline{\mathfrak{t}} = qg(K)+p-1$, and vanishes outside this range.

ABSTRACT

We establish surgery formulas for filtration of the Heegaard Floer homology associated with p/q surgery on a null-homologous knot K in a three-manifold Y, induced by K_{p/q}. Here K_{p/q} is the core of the attached solid torus (which produces the surgery). This would generalize a result of Ozsvath and Szabo. We will re-prove that surgery on non-trivial knots can not produce S^3, as a corollary of a non-vanishing result for the knot Floer homology of K_{p/q}.

Motivation & Objective

• To generalize Ozsváth and Szabó's surgery formula for integer surgeries to rational surgeries on null-homologous knots in homology spheres.
• To compute the filtration on the Heegaard Floer complex induced by the core of the surgery torus in $Y_{p/q}(K)$, extending the framework of [OS4].
• To re-prove that nontrivial knot surgeries cannot yield $S^3$, using non-vanishing results in knot Floer homology.
• To provide a computational tool for the filtered chain complex of the surgered manifold via a mapping cone construction involving $\mathbb{Z}[ζ]/(\zeta^q=1)$-twisted maps.

Proposed method

• Construct a filtered chain complex $\mathbb{D}$ from the $\text{CFK}^\infty(Y,K)$ complex, using four integer gradings and a differential $\partial_\mathbb{D}$ derived from $\partial^\infty$.
• Define subcomplexes $\mathbb{D}_\delta$ based on the invariant $\Delta(a) = i-j+k-l$, and construct $\mathbb{D}^{up}$ and $\mathbb{D}^{down}$ as copies of $\mathbb{D}_1$ with distinct $\mathbb{Z}\oplus\mathbb{Z}$ gradings.
• Introduce a chain map $g_{p/q}$ from $\mathbb{D}^{up} \otimes \mathbb{Z}[\zeta]/(\zeta^q=1)$ to $\mathbb{D}^{down} \otimes \mathbb{Z}[\zeta]/(\zeta^q=1)$ using a shift involving $\lfloor (t+p)/q \rfloor$, encoding the surgery data.
• Define $\overline{f}_{p/q} = \text{Id} + g_{p/q}$ and form the mapping cone $\mathbb{M}(\overline{f}_{p/q})$, which carries the filtered structure of the surgered manifold's homology.
• Use homotopy equivalence $\tau$ between filtrations from different marked points to ensure invariance under choice of basepoint.
• Identify the homology of the mapping cone with $\widehat{\text{HFK}}(K_r, \underline{\mathfrak{t}})$ via decomposition into positive test domains and analysis of generators in $C\{i\leq 0,k=0\}$ and $C\{i=0,k\leq 0\}$.

Experimental results

Research questions

• RQ1 How does the filtration on the Heegaard Floer complex transform under rational surgery on a null-homologous knot in a homology sphere?
• RQ2 What is the precise range of $\underline{\mathfrak{t}} \in \mathbb{Z}$ for which $\widehat{\text{HFK}}(K_r, \underline{\mathfrak{t}})$ is non-zero after $p/q$-surgery on a knot $K \subset S^3$?
• RQ3 Can the non-vanishing of $\widehat{\text{HFK}}(K_r, \underline{\mathfrak{t}})$ be used to obstruct rational surgeries from yielding $S^3$?
• RQ4 How does the mapping cone construction with $\mathbb{Z}[\zeta]/(\zeta^q=1)$-twisted maps capture the filtered chain complex of the surgered manifold?

Key findings

• For a knot $K \subset S^3$ of genus $g(K)$, the knot Floer homology $\widehat{\text{HFK}}(K_r, \underline{\mathfrak{t}})$ is non-zero precisely at $\underline{\mathfrak{t}} = -qg(K)$ and $\underline{\mathfrak{t}} = qg(K)+p-1$, with $\widehat{\text{HFK}}(K_r, -qg(K)) \cong \widehat{\text{HFK}}(K, g(K)) \neq 0$.
• For all $\underline{\mathfrak{t}} < -qg(K)$ or $\underline{\mathfrak{t}} \geq qg(K)+p$, the homology $\widehat{\text{HFK}}(K_r, \underline{\mathfrak{t}})$ vanishes, as shown by triviality of the corresponding homology groups in the mapping cone decomposition.
• The non-vanishing of $\widehat{\text{HFK}}(K_r, \underline{\mathfrak{t}})$ at the extremal $\underline{\mathfrak{t}}$-values implies that no nontrivial knot surgery can produce $S^3$, as such a surgery would require $\widehat{\text{HFK}}(S^3, \underline{\mathfrak{t}}) = 0$ for all $\underline{\mathfrak{t}} \neq 0$, contradicting the non-vanishing result.
• The complex $\mathbb{M}(\overline{f}_{p/q})$ computes the filtered $\text{CFK}^\infty$ of $Y_{p/q}(K)$, with the ${\mathrm{Spin}}^c$ structure assigned via $\underline{\mathfrak{s}} = q(i(\mathbf{x})+j-k) + p(i-j) + t$.
• The isomorphism $\widehat{\text{HFK}}(K_r, -qg(K)) \cong \widehat{\text{HFK}}(K, g(K)) \neq 0$ is established via identification of $H_*(\widehat{\mathbb{A}}_0(-qg(K)))$ with $C\{i=0,k\leq 0\}(-g(K))$, where generators with $i(\mathbf{x}) < -g(K)$ cancel in homology.
• Similarly, $\widehat{\text{HFK}}(K_r, qg(K)+p-1) \cong \widehat{\text{HFK}}(K, g(K)) \neq 0$ is shown by analyzing $C\{i\leq 0,k=0\}(g(K)-1)$ and showing only $i(\mathbf{x}) = g(K)$ generators survive in homology.