A remark on the geography problem in Heegaard Floer homology

TL;DR

This paper introduces new obstructions to module structures in Heegaard Floer homology, characterizes possible modules for integer homology spheres with minimal reduced Floer homology, and applies these results to knot invariants.

Contribution

It provides novel obstructions to module structures in Heegaard Floer homology and characterizes modules for certain homology spheres, impacting the understanding of knot invariants.

Findings

Only two modules arise as the Heegaard Floer homology of certain homology spheres.

The chain complex used by Ozsváth, Stipsicz, and Szabó cannot be realized as a knot Floer complex.

New obstructions to module structures in Heegaard Floer homology are established.

Abstract

We give new obstructions to the module structures arising in Heegaard Floer homology. As a corollary, we characterize the possible modules arising as the Heegaard Floer homology of an integer homology sphere with one-dimensional reduced Floer homology. Up to absolute grading shifts, there are only two. We use this corollary to show that the chain complex depicted by Ozsv\'ath, Stipsicz, and Szab\'o to argue that there is no algebraic obstruction to the existence of knots with trivial $\epsilon$ invariant and non-trivial $\Upsilon$ invariant cannot be realized as the knot Floer complex of a knot.