Non-separating spheres and twisted Heegaard Floer homology

TL;DR

The paper demonstrates that the presence of a non-separating sphere in a 3-manifold implies certain twisted Heegaard Floer homologies vanish, leading to new insights into Dehn surgery on knots in these manifolds.

Contribution

It establishes a link between non-separating spheres and vanishing twisted Heegaard Floer homology, extending results known for knots in L-spaces.

Findings

Vanishing of some twisted Heegaard Floer homology in manifolds with non-separating spheres

Applications to Dehn surgery results on knots in such manifolds

Extension of results previously known for knots in L-spaces

Abstract

If a 3--manifold $Y$ contains a non-separating sphere, then some twisted Heegaard Floer homology of $Y$ is zero. This simple fact allows us to prove several results about Dehn surgery on knots in such manifolds. Similar results have been proved for knots in $L$--spaces.