The contact invariant in sutured Floer homology

Journal of Differential Geometry

We present an alternate description of the Ozsváth-Szabó contact class in Heegaard Floer homology. Using our contact class, we prove that if a contact structure (M, ξ) has an adapted open book decomposition whose page S is a once-punctured torus, then the monodromy is rightveering if and only if the contact structure is tight.

arXiv: Geometric Topology, 2017

Seiberg-Witten (Floer) theory, Ozsvath-Szabo's Heegaard Floer theory, Hutchings's embedded contact homology, in different stages of development, define (or are expected to define) packages of invariants for 3- and 4-manifolds (including manifolds with boundary and manifolds with certain types of corners). We describe what are known about their relationship, what are expected, and raise some questions along the way.

arXiv (Cornell University), 2016

We define an invariant of contact structures in dimension three from Heegaard Floer homology. This invariant takes values in the set Zě0 Y t8u. It is zero for overtwisted contact structures, 8 for Stein fillable contact structures, non-decreasing under Legendrian surgery, and computable from any supporting open book decomposition. As an application, we obstruct Stein fillability on contact 3-manifolds with non-vanishing Ozsváth-Szabó contact class.

Topology and its Applications, 2016

In this note, we exhibit infinite families of tight non-fillable contact manifolds supported by planar open books with vanishing Heegaard Floer contact invariants. Moreover, we also exhibit an infinite such family where the supported manifold is hyperbolic.

2008

We describe the natural gluing map on sutured Floer homology which is induced by the inclusion of one sutured manifold (M',\Gamma') into a larger sutured manifold (M,\Gamma), together with a contact structure on M-M'. As an application of this gluing map, we produce a (1+1)-dimensional TQFT by dimensional reduction and study its properties.

arXiv (Cornell University), 2020

We compute the homotopy type of the space of embeddings of convex disks with Legendrian boundary into a tight contact 3-manifold, whenever the sum of the absolute value of the rotation number of the boundary with the Thurston-Bennequin invariant is −1, proving that it is homotopy equivalent to the space of smooth embeddings. Using the same ideas it is also determined the homotopy type of the space of embeddings of convex spheres into a tight 3-fold in terms of the space of smooth spheres. As a consequence we determine the homotopy type of the space of long Legendrian unknots, satisfying the previous condition, into a tight 3-fold and also of the space of long transverse unknots with self-linking number −1, proving that these spaces are homotopy equivalent to the space of smooth long unknots. We also determine the homotopy type of the contactomorphism group of every universally tight handlebody, the standard S 1 ×S 2 and every Legendrian fibration over a compact orientable surface with non-empty boundary, partially solving a conjecture due to E. Giroux. Finally, we show that the space of embeddings of Legendrian (n, n)torus links with maximal Thurston-Bennequin invariant is homotopy equivalent to U(2) × K(M n , 1), where M n is the mapping class group of the 2-sphere with n-holes. Contents 1. Introduction. 1 2. Preliminaries. 11 3. Eliashberg-Mishachev and Giroux Theorems. 16 4. The Legendrian unknot in a tight contact 3-manifold. 28 5. Applications. 31 References 36 1. Introduction. Throughout this article M will be an oriented compact 3-manifold and ξ a positive contact structure on M , that we will always assume cooriented. The pair (M, ξ) is a contact 3-manifold. A remarkable fact, due to Gray [Gra59], is that there are no

2010

Let M be a closed, connected and oriented 3-manifold. This article is the first of a five part series that constructs an isomorphism between the Heegaard Floer homology groups of M and the corresponding Seiberg-Witten Floer homology groups of M.

2021

This paper presents, with explanatory details, the handle decompositions, fundamental groups and homology groups of 3-manifolds, including some knot complements. Hence, along this paper, when the word manifold appears it is implicit that its dimension is 3, except when explicitly generalized for n dimensions, n N. The results were obtained for: 3-torus (T = S S S), projective space P , trefoil (31), figure-eight (41), cinquefoil (51) and three-twist (52).

Memoirs of the American Mathematical Society

We define combinatorial Floer homology of a transverse pair of noncontractibe nonisotopic embedded loops in an oriented 2-manifold without boundary, prove that it is invariant under isotopy, and prove that it is isomorphic to the original Lagrangian Floer homology.

Journal of Fixed Point Theory and Applications, 2008

We define the Floer complex for Hamiltonian orbits on the cotangent bundle of a compact manifold which satisfy non-local conormal boundary conditions. We prove that the homology of this chain complex is isomorphic to the singular homology of the natural path space associated to the boundary conditions.