Lattice cohomology of normal surface singularities

TL;DR

This paper introduces a new graded Z[U]-module associated with negative definite plumbed 3-manifolds, linking topological invariants with complex singularity theory and conjecturally relating to Heegaard-Floer homology.

Contribution

It constructs a novel algebraic invariant from plumbed graphs that encodes topological and analytic information, extending the understanding of surface singularities and 3-manifold invariants.

Findings

The module matches Heegaard-Floer homology for rational homology spheres (conjectural).

It relates the Euler characteristic to analytic invariants of singularities.

Provides insights into the Seiberg--Witten Invariant Conjecture.

Abstract

For any negative definite plumbed 3-manifold M we construct from its plumbed graph a graded Z[U]-module. This, for rational homology spheres, conjecturally equals the Heegaard-Floer homology of Ozsvath and Szabo, but it has even more structure. If M is a complex singularity link then the normalized Euler-characteristic can be compared with the analytic invariants. The Seiberg--Witten Invariant Conjecture is discussed in the light of this new object.