Lattice Cohomology Research Articles

Lattice Cohomology of Partially Ordered Sets

In this paper we introduce a construction for a weighted CW complex (and the associated lattice cohomology) corresponding to partially ordered sets with some additional structure. This is a generalization of the construction seen in [4] where we started from a system of subspaces of a given vector space. We then proceed to prove some basic properties of this construction that are in many ways analogous to those seen in the case of subspaces, but some aspects of the construction result in complexities not present in that scenario.

Categorification of the plurigenera of Gorenstein normal surface singularities

Consider a complex normal surface singularity and its three plurigenera, the m-th L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}–plurigenus of Watanabe, the m-th plurigenus of Knöller and the m-th log-plurigenus of Morales. For any of these invariants we construct a double graded Z[U]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}[U]$$\\end{document}–module, whose Euler characteristic is the chosen plurigenus. The three outputs are compared with the analytic lattice cohomology of the germ, whose Euler characteristic is the classical geometric genus.

Lattice cohomology and q-series invariants of 3-manifolds

Abstract In this paper, an invariant is introduced for negative definite plumbed 3-manifolds equipped with a spin c {{}^{c}} -structure. It unifies and extends two theories with rather different origins and structures. One theory is lattice cohomology, motivated by the study of normal surface singularities, known to be isomorphic to the Heegaard Floer homology for certain classes of plumbed 3-manifolds. Another specialization gives BPS q-series which satisfy some remarkable modularity properties and recover SU ⁢ ( 2 ) {{\rm SU}(2)} quantum invariants of 3-manifolds at roots of unity. In particular, our work gives rise to a 2-variable refinement of the Z ^ {\widehat{Z}} -invariant.

Heegaard Floer invariants of contact structures on links of surface singularities

Let a contact 3-manifold (Y, \xi_0) be the link of a normal surface singularity equipped with its canonical contact structure \xi_0 . We prove a special property of such contact 3-manifolds of "algebraic" origin: the Heegaard Floer invariant c^+(\xi_0)\in \mathrm {HF}^+(-Y) cannot lie in the image of the U -action on \mathrm {HF}^+(-Y) . It follows that Karakurt's "height of U -tower" invariants are always 0 for canonical contact structures on singularity links, which contrasts the fact that the height of U -tower can be arbitrary for general fillable contact structures. Our proof uses the interplay between the Heegaard Floer homology and Némethi's lattice cohomology.

The geometric genus of hypersurface singularities

Using the path lattice cohomology we provide a conceptual topological characterization of the geometric genus for certain complex normal surface singularities with rational homology sphere links, which is uniformly valid for all superisolated and Newton non-degenerate hypersurface singularities.

Lattice cohomology and rational cuspidal curves

We show a counterexample to a conjecture of de Bobadilla, Luengo, Melle-Hernandez and Nemethi on rational cuspidal projective plane curves. The counterexample is a tricuspidal curve of degree 8. On the other hand, we show that if the number of cusps is at most 2, then the original conjecture can be deduced from the recent results of Borodzik and Livingston and (lattice cohomology) computations of Nemethi and Roman. Then we formulate a `simplified' (slightly weaker) version, more in the spirit of the motivation of the original conjecture (comparing index type numerical invariants), and we prove it for all currently known rational cuspidal curves. We make all these identities and inequalities more transparent in the language of lattice cohomology of surgery 3-manifolds $S^3_{-d}(K)$, where $K$ is a connected sum of algebraic knots $\{K_i\}_i$. Finally, we prove that the zeroth lattice cohomology of this surgery manifold depends only on the multiset of multiplicities occurring in the multiplicity sequences describing the algebraic knots $K_i$. This result is closely related to the lattice-cohomological reformulation of the above mentioned theorems and conjectures, and provides new computational and comparison procedures.

Reduction Theorem for Lattice Cohomology

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 |${\mathbb Z}^s$|⁠, 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. 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 |${\mathbb H}^q=0$| of the lattice cohomology for q≥ν, where ν is the number of “bad” vertices. (This bound is sharp.)

A surgery triangle for lattice cohomology

Lattice cohomology, defined by N\'emethi in (arXiv:0709.0841), is an invariant of negative definite plumbed 3-manifolds which conjecturally computes the Heegaard Floer homology HF^+. We prove a surgery exact triangle for the lattice cohomology analogous to the one for HF^+. This is a step towards comparing these two invariants.

The Seiberg–Witten invariants of negative definite plumbed 3-manifolds

Assume that \Gamma is a connected negative definite plumbing graph, and that the associated plumbed 3-manifold M is a rational homology sphere. We provide two new combinatorial formulae for the Seiberg–Witten invariant of M . The first one is the constant term of a 'multivariable Hilbert polynomial', it reflects in a conceptual way the structure of the graph \Gamma , and emphasizes the subtle parallelism between these topological invariants and the analytic invariants of normal surface singularities. The second formula realizes the Seiberg–Witten invariant as the normalized Euler characteristic of the lattice cohomology associated with \Gamma , supporting the conjectural connections between the Seiberg–Witten Floer homology, or the Heegaard–Floer homology, and the lattice cohomology.

Lattice Cohomology of Normal Surface Singularities

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.

Bounded cohomology of lattices in higher rank Lie groups

We prove that the natural map H^2_b(\Gamma)\to H^2(\Gamma) from bounded to usual cohomology is injective if \Gamma is an irreducible cocompact lattice in a higher rank Lie group. This result holds also for nontrivial unitary coefficients, and implies finiteness results for \Gamma : the stable commutator length vanishes and any C^1 -action on the circle is almost trivial. We introduce the continuous bounded cohomology of a locally compact group and prove our statements by relating H^\bullet_b(\Gamma) to the continuous bounded cohomology of the ambient group with coefficients in some induction module.

On the Cohomology of Lattices in Solvable Lie Groups

On the Cohomology of Lattices in Solvable Lie Groups