Calculating Heegaard-Floer Homology by Counting Lattice Points in Tetrahedra

TL;DR

This paper introduces a new way to compute Heegaard-Floer homology for Seifert homology spheres by counting lattice points in tetrahedra, establishing a link with Casson invariant and classifying certain manifolds.

Contribution

It defines a complexity measure for Seifert homology spheres via lattice point counting, connecting it to existing invariants and enabling classification of specific manifold types.

Findings

Complexity is equivalent to a version of Casson invariant.

Finitely many Seifert homology spheres have prescribed Heegaard-Floer homology.

All Seifert homology spheres up to complexity two are listed.

Abstract

We introduce a notion of complexity for Sefiert homology spheres by establishing a correspondence between lattice point counting in tethrahedra and the Heegaard-Floer homology. This complexity turns out to be equivalent to a version of Casson invariant and it is monotone under a natural partial order in the set of Seifert homology spheres. Using this interpretation we prove that there are finitely many Seifert homology spheres with prescribed Heegaard-Floer homology. As an application, we characterize L-spaces and weakly elliptic manifolds among Seifert homology spheres. Also, we list all the Seifert homology spheres up to complexity two.