Graph cobordisms and Heegaard Floer homology

TL;DR

This paper develops a graph TQFT for Heegaard Floer homology that extends previous theories to disconnected cobordisms and provides explicit formulas for the fundamental group action, revealing triviality in some cases and non-triviality in others.

Contribution

It introduces a new graph TQFT for Heegaard Floer homology that generalizes existing theories to disconnected cobordisms and computes the fundamental group action explicitly.

Findings

The $ ext{pi}_1$-action is trivial on homology for plus, minus, and infinity flavors.

Explicit formula for the chain homotopy type of the $ ext{pi}_1$-action.

Examples show non-trivial $ ext{pi}_1$-action on the hat flavor.

Abstract

We construct a graph TQFT for the minus flavor of Heegaard Floer homology. Our graph TQFT extends Ozsv\'{a}th and Szab\'{o}'s TQFT for closed and connected 3-manifolds, and allows for cobordisms with disconnected ends. As an application, we give an explicit formula for the chain homotopy type of the $\pi_1$-action on Heegaard Floer homology. We show that on homology the $\pi_1$-action is trivial on the plus, minus and infinity flavors, but give examples where it is non-trivial on the hat flavor.