Reduction theorem for lattice cohomology

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.)