A remark on the geography problem in Heegaard Floer homology

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.