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An Excision Theorem in Heegaard Floer Theory

Neda Bagherifard

Abstract

Let $Y_1$ be a closed, oriented 3-manifold and $Σ$ denote a non-separating closed, orientable surface in $Y_1$ which consists of two connected components of the same genus. By cutting $Y_1$ along $Σ$ and re-gluing it using an orientation-preserving diffeomorphism of $Σ$ we obtain another closed, oriented 3-manifold $Y_2$. When the excision surface $Σ$ is of genus one, we show that twisted Heegaard Floer homology groups of $Y_1$ and $Y_2$ (twisted with coefficients in the universal Novikov ring) are isomorphic. We use this excision theorem to demonstrate that certain manifolds are not related by the excision construction on a genus one surface. Additionally, we apply the excision formula to compute twisted Heegaard Floer homology groups of 0-surgery on certain two-component links, including some families of 2-bridge links.

An Excision Theorem in Heegaard Floer Theory

Abstract

Let be a closed, oriented 3-manifold and denote a non-separating closed, orientable surface in which consists of two connected components of the same genus. By cutting along and re-gluing it using an orientation-preserving diffeomorphism of we obtain another closed, oriented 3-manifold . When the excision surface is of genus one, we show that twisted Heegaard Floer homology groups of and (twisted with coefficients in the universal Novikov ring) are isomorphic. We use this excision theorem to demonstrate that certain manifolds are not related by the excision construction on a genus one surface. Additionally, we apply the excision formula to compute twisted Heegaard Floer homology groups of 0-surgery on certain two-component links, including some families of 2-bridge links.

Paper Structure

This paper contains 8 sections, 20 theorems, 141 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Background on twisted Heegaard Floer theory
  3. Twisted Heegaard Floer homology groups
  4. $\omega$-Twisted graph cobordisms
  5. Absolute grading and surgery exact sequence
  6. Sums and compositions of twisted cobordism maps
  7. Proof of Theorem \ref{['MainThm']}
  8. Applications

Key Result

Theorem 1.1

phdthesis Let $Y_2$ be constructed from $Y_1$ as explained above. Denote the excision surface in $Y_1$ and its corresponding copy in $Y_2$ by $F=\Sigma_1\cup\Sigma_2$ where $g(\Sigma_i)\geq2$, $i=1,2$. Then Moreover, this isomorphism and its inverse are induced by restricted graph cobordisms. Here $HF(Y,\mathfrak{s})$ denotes $HF_{red}(Y,\mathfrak{s})=HF^+(Y,\mathfrak{s})\cong\mathbf{HF}^-(Y,\mat

Figures (16)

  • Figure 1: The construction of excision when $Y_1$ is connected.
  • Figure 2: The doubly pointed Heegaard diagram $\mathcal{H}_0$ used in the definition of free stabilization maps. The Heegaard surface is a sphere.
  • Figure 3: This is Figure A.4 of phdthesis.
  • Figure 4: This is Figure 14.1 of Z15.
  • Figure 5: This is Figure 14.2 of Z15.
  • ...and 11 more figures

Theorems & Definitions (46)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • ...and 36 more