Two Applications of Twisted Floer Homology

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.

Manifolds with small Heegaard Floer ranks

“…In addition to the Thurston norm of a homology class, α ∈ H 2 (Y, ∂Y ), the Ozsváth-Szabó invariants answer the subtler question of when a knot complement or closed three-manifold fibers over the circle with fiber an embedded surface, Σ, whose homology class equals α [8,20,21,1]. For knots, this again has a beautiful corollary, namely that knot Floer homology detects the trefoil and figure eight knots [8].…”

Nonseparating spheres and twisted Heegaard Floer homology

“…Moreover, using Heegaard Floer homology, we can show that if HF red (Y ) = 0 then K has Property G. (For Property G2, the proof can be found in [10,1]. The proof for Property G1 is similar.…”

Some applications of Gabaiʼs internal hierarchy

“…This case is sufficient for our applications. For our purpose, we will strengthen the statement of Theorem 1.1 by introducing 1 We will not prove this fact here, since we do not need it in our paper. 2 For example, let M = S 1 × B, where B is an orientable surface with exactly two boundary components C + , C − .…”