The twisted Floer homology of torus bundles

Abstract

We prove an exact sequence for ! -twisted Heegaard Floer homology. As a corollary, given a torus bundle Y over the circle and a cohomology class OE! 2 H 2 .Y I Z/ which evaluates nontrivially on the fiber, we compute the Heegaard Floer homology of Y with twisted coefficients in the universal Novikov ring.

Two Applications of Twisted Floer Homology

“…In Section 4, we give an proof of Theorem 1.2 following Ghiggini's argument. In Section 5, we prove Theorem 1.4 by using Theorem 1.1 and the exact sequence for ω-twisted Floer homology from [1].…”

“…The first author would also like to thank Thomas Peters for many helpful discussions during their joint work [1].…”

“…A special twisted Floer homology is used to investigate torus bundles over the circle in [1]. Consider the universal Novikov ring [11, Section 11.1] Λ = r∈R a r t r a r ∈ R, #{a r |a r = 0, r ≤ c} < ∞ for any c ∈ R .…”

“…Ozsváth and Szabó suggested us that one may use Heegaard Floer homology with twisted coefficients in some Novikov ring. Some calculations for torus bundles has been done in an earlier paper [1] along this line.…”

“…In [1] this twisted Heegaard Floer homology is calculated for torus bundles. The above theorem completes in some sense a result of Ozsváth and Szabó [16] which states that a surface bundle over circle has monic Heegaard Floer homology.…”