Reduction Theorem for Lattice Cohomology

Abstract

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, whenever 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, or, how far is M from an L-space.)The effect of the reduction appears also at the level of certain multivariable (topological Poincaré) 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 H q = 0 of the lattice cohomology for q ≥ ν, where ν is the number of 'bad' vertices. (This bound is sharp.)

Calculating Heegaard–Floer homology by counting lattice points in tetrahedra

“…Recently László and Némethi [8] developed a version of Erhart theory for Seiberg-Witten invariants of plumbed rational homology sphere. It would be interesting to compare the lattice point counting argument there and the one given above.…”

“…We would like to point out that the complexity mentioned above has already been studied as the normalized Seiberg-Witten invariant, or as the normalized Euler characteristic of Heegaard-Floer homology, in several places including [13], [11], [8], [7]. It is the topological candidate for geometric genus of certain analytical singularities whose links are rational homology spheres.…”

On the geometry of strongly flat semigroups and their generalizations

“…However, it will turn out that the tools are far to be merely arithmetical, they rely on the mathematical machinery of 'generalized Laufer computation sequences', which originally was used in the computation of analytic invariants of surface singularities, and later in the computation of the lattice cohomology of their links. In order to make the presentation more complete, in the following we review some motivations, definitions and facts from [N05,LN15] about the lattice cohomology and then we present the special computation sequences needed in the proof. Heegaard-Floer homology by Ozsváth and Szabó [OSz04] is one of the most important and highlighted invariants of 3-manifolds.…”

“…The Reduction theorem of [LN15] allows to reduce the rank of the lattice and evidences the complexity of the cohomology. In fact, in the case of Seifert rational homology spheres (and in general, in the case of plumbed manifolds associated with 'almost rational' graphs) the higher cohomology…”

Surgery formulae for the Seiberg–Witten invariant of plumbed 3-manifolds

“…We recall that a collection of vertices I of V is called 'bad' if by decreasing the decoration of these vertices on the graph we obtain a rational graph (cf. [N05,LN15] (2) A special family of graph manifolds when T \ I are all rational is provided by S 3 −d (K), the (−d)-surgery along the connected sum K = K 1 # . .…”