Lattice Cohomology of Normal Surface Singularities

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 Ozsváth and Szabó, 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 of [16], [13] is discussed in the light of this new object. §1. IntroductionThe article is a symbiosis of singularity theory and low-dimensional topology. Accordingly, it is preferable to separate its goals in two categories.From the point of view of 3-dimensional topology, the article contains the following main result. For every negative definite plumbed 3-manifold it constructs a graded Z[U ]-module from the combinatorics of the plumbing graph. This for rational homology spheres conjecturally equals the Heegaard-Floer homology of Ozsváth and Szabó. In fact, it has more structure (e.g. instead of a

Seiberg–Witten Invariant of the Universal Abelian Cover of $${S_{-p/q}^{3}(K)}$$

“…[15,17,20] If G is a (not necessarily minimal) graph of S 3 or of a lens-space then s h (G) = 0 for every h ∈ H.…”

“…In this article we will involve another cohomology theory with similar property. Since S 3 −p/q (K) is representable by a negative definite plumbing graph, via [21] we view the SW invariants as Euler characteristics of lattice cohomologies introduced in [17]. The big advantage of the lattice cohomology over the classical definition of Heegaard-Floer homology is that it is computable algorithmically from the plumbing graph.…”

“…The normalized SW invariant s h (G) can also be expressed as the Euler characteristic of the lattice cohomology. The advantage of this approach is that it provides an alternative, completely elementary way to define s, as the definition of the lattice cohomology is purely combinatorial from the plumbing graph G. We briefly recall the definition and some facts about the lattice cohomology associated with a QHS 3 3-manifold with negative definite plumbing graph G. For more see [17,26].…”

Reduction Theorem for Lattice Cohomology

“…The second author in [11,16] associated with such an M (and any fixed spin c -structure s of M ) a graded Z[U ]-module H * (M, s), called the lattice cohomology of M . The construction was strongly influenced by the Artin-Laufer program of normal surface singularities (targeting topological characterization of certain analytic invariants), cf.…”

“…The construction was strongly influenced by the Artin-Laufer program of normal surface singularities (targeting topological characterization of certain analytic invariants), cf. [20,11,16], and by the work of Ozsváth and Szabó on Heegaard-Floer theory, especially [26] (see also their long list of papers in the subject).…”