Contact structures on AR-singularity links

Abstract

An isolated complex surface singularity induces a canonical contact structure on its link. In this paper, we initiate the study of the existence problem of Stein cobordisms between these contact structures depending on the properties of singularities. As a first step, we construct an explicit Stein cobordism from any contact 3-manifold to the canonical contact structure of a proper almost rational singularity introduced by Némethi. We also show that the construction cannot always work in the reverse direction: in fact, the U-filtration depth of contact Ozsváth–Szabó invariant obstructs the existence of a Stein cobordism from a proper almost rational singularity to a rational one. Along the way, we detect the contact Ozsváth–Szabó invariants of those contact structures fillable by an AR plumbing graph, generalizing an earlier work of the first author.

Analytic lattice cohomology of surface singularities

“…In such cases the use of graded roots is significantly more convenient than any other method, see e.g. [18,23,24,25].…”

Analytic lattice cohomology of surface singularities, II (the equivariant case)

“…In such cases the use of graded roots is significantly more convenient than any other method, see e.g. [14,17,18,19]. 1.2.…”

Heegaard Floer invariants of contact structures on links of surface singularities

“…Karakurt computes ht for a number of contact structures obtained by Legendrian surgery, and shows that ht can take arbitrary integer values from 0 to C1. In [8], Karakurt and Öztürk show that the height of tower is 0 for canonical contact structures on links of "almost rational" (AR) singularities, using the fact that Heegaard Floer homology is isomorphic to Némethi's lattice cohomology [11,12] for 3-manifolds of this type. For rational singularities, it is easy to see that ht D C1 for every contact structure on the link: this follows from the fact that the link Y of a rational singularity is an L-space, i.e.…”

“…Y; s/ D T for every Spin c -structure s on Y , see [16,11]. Karakurt and Öztürk ask whether height of tower can take arbitrary integer values for canonical contact structures on links of general normal surface singularities [8,Question 6.2]. It follows immediately from Theorem 1.2 that the answer is manifestly no: Corollary 1.3.…”

“…Similarly to [8], our proof also uses the interplay between Heegaard Floer homology and lattice cohomology of [11,12]. Lattice cohomology is defined in a combinatorial way, using the intersection lattice of the plumbing graph (the dual resolution graph of the singularity).…”