Two applications of twisted Floer homology

TL;DR

This paper explores two applications of twisted Floer homology: identifying torus bundle structures in 3-manifolds and characterizing fibered knots in L-spaces, advancing understanding in low-dimensional topology.

Contribution

It proves that twisted Heegaard Floer homology detects torus bundle structures and characterizes fibered knots in L-spaces, filling gaps in previous results.

Findings

Twisted Floer homology determines if a 3-manifold is a torus bundle.

In L-spaces, if 0-surgery on a genus 1 null-homologous knot yields a fibered manifold, then the knot is fibered.

Addresses missing cases in earlier topological classifications.

Abstract

Given an irreducible closed 3--manifold $Y$, we show that its twisted Heegaard Floer homology determines whether $Y$ is a torus bundle over the circle. Another result we will prove is, if $K$ is a genus 1 null-homologous knot in an $L$--space, and the 0--surgery on $K$ is fibered, then $K$ itself is fibered. These two results are the missing cases of earlier results due to the second author.