On the twisted Floer homology of mapping tori of periodic diffeomorphisms

TL;DR

This paper develops a formula to compute twisted Floer homology for certain 3-manifolds obtained via surgery and applies it to analyze the Floer homology of mapping tori of periodic surface diffeomorphisms, linking it to symplectic Floer homology.

Contribution

It introduces a new formula connecting twisted Floer homology of surgered manifolds with twisted knot Floer homology and computes the Floer homology of mapping tori of periodic diffeomorphisms.

Findings

Computed the twisted Heegaard Floer homology of mapping tori of periodic diffeomorphisms.

Established that in the second-to-highest Spin^c structure, the Floer homology is a free module with rank equal to the Lefschetz number.

Confirmed that after mod 2 reduction, the Floer homology matches the symplectic Floer homology as calculated by Gautschi.

Abstract

Let K \subset Y be a knot in a three manifold which admits a longitude-framed surgery such that the surgered manifold has first Betti number greater than that of Y. We find a formula which computes the twisted Floer homology of the surgered manifold, in terms of twisted knot Floer homology. Using this, we compute the twisted Heegaard Floer homology \underline{HF}^+ of the mapping torus of a diffeomorphism of a closed Riemann surface whose mapping class is periodic, giving an almost complete description of the structure of these groups. When the mapping class is nontrivial, we find in particular that in the "second-to-highest" Spin^c structure, this is isomorphic to a free module (over a certain ring) whose rank is equal to the Lefschetz number of the diffeomorphism. After taking a tensor product with Z/2Z, this agrees precisely with the symplectic Floer homology of the diffeomorphism, as calculated by Gautschi.