Reduction theorem for lattice cohomology

TL;DR

This paper introduces a reduction theorem for lattice cohomology of plumbed 3-manifolds, simplifying computations by focusing on 'bad' vertices, and provides new formulas for Seiberg--Witten invariants.

Contribution

It reduces the rank of lattice cohomology to the number of 'bad' vertices, streamlining calculations and linking topological invariants to graph properties.

Findings

Reduction of lattice cohomology rank to 'bad' vertices

New formulas for Seiberg--Witten invariants

Vanishing of lattice cohomology for degrees above number of 'bad' vertices

Abstract

The lattice cohomology of a plumbed 3--manifold $M$ associated with a connected negative definite plumbing graph is an important tool in the study of topological properties of $M$, and in the comparison of the topological properties with analytic ones when $M$ is realized as complex analytic singularity link. By definition, its computation is based on the (Riemann--Roch) weights of the lattice points of $\Z^s$, where $s$ is the number of vertices of the plumbing graph. The present article reduces the rank of this lattice to the number of `bad' vertices of the graph. (Usually the geometry/topology of $M$ is codified exactly by these `bad' vertices via surgery or other constructions. Their number measures how far is the plumbing graph from a rational one.) The effect of the reduction appears also at the level of certain multivariable (topological Poincar\'e) series as well. Since from these series one can also read the Seiberg--Witten invariants, the reduction theorem provides new formulae for these invariants too. The reduction also implies the vanishing $\bH^q=0$ of the lattice cohomology for $q\geq \nu$, where $\nu$ is the number of `bad' vertices. (This bound is sharp.)