An Excision Theorem in Heegaard Floer Theory

TL;DR

This paper proves an excision theorem in Heegaard Floer theory for genus one surfaces, showing invariance of twisted homology under certain cut-and-paste operations, and applies it to compute homologies of specific 3-manifolds.

Contribution

It establishes a new excision theorem for twisted Heegaard Floer homology in genus one cases, expanding computational tools and understanding of 3-manifold invariants.

Findings

Twisted Heegaard Floer homology groups are invariant under genus one excision.

Certain manifolds are shown not to be related by genus one excision.

Computed twisted homology groups for 0-surgery on specific links.

Abstract

Let $Y_1$ be a closed, oriented 3-manifold and $\Sigma$ 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 $\Sigma$ and re-gluing it using an orientation-preserving diffeomorphism of $\Sigma$ we obtain another closed, oriented 3-manifold $Y_2$. When the excision surface $\Sigma$ 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.