Robert Lipshitz

Bordered Heegaard Floer homology

We construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with parametrized boundary, the invariant comes in two different versions, one of which (type D) is a module over the algebra and the other of which (type A) is an A-infinity module. Both are well-defined up to chain homotopy equivalence. For a decomposition of a 3-manifold into two pieces, the A-infinity tensor product of the type D module of one piece and the type A module from the other piece is HF^ of the glued manifold. As a special case of the construction, we specialize to the case of three-manifolds with torus boundary. This case can be used to give another proof of the surgery exact triangle for HF^. We relate the bordered Floer homology of a three-manifold with torus boundary with the knot Floer homology of a filling.

A cylindrical reformulation of Heegaard Floer homology

We reformulate Heegaard Floer homology in terms of holomorphic curves in the cylindrical manifold U a0;1c R, where U is the Heegaard surface, instead of Sym g .U/. We then show that the entire invariance proof can be carried out in our setting. In the process, we derive a new formula for the index of the @‐operator in Heegaard Floer homology, and shorten several proofs. After proving invariance, we show that our construction is equivalent to the original construction of Ozsvath‐Szabo. We conclude with a discussion of elaborations of Heegaard Floer homology suggested by our construction, as well as a brief discussion of the relation with a program of C Taubes.

Bimodules in bordered Heegaard Floer homology

Bordered Heegaard Floer homology is a three-manifold invariant which associates to a surface F an algebra A(F) and to a three-manifold Y with boundary identified with F a module over A(F). In this paper, we establish naturality properties of this invariant. Changing the diffeomorphism between F and the boundary of Y tensors the bordered invariant with a suitable bimodule over A(F). These bimodules give an action of a suitably based mapping class group on the category of modules over A(F). The Hochschild homology of such a bimodule is identified with the knot Floer homology of the associated open book decomposition. In the course of establishing these results, we also calculate the homology of A(F). We also prove a duality theorem relating the two versions of the 3-manifold invariant. Finally, in the case of a genus one surface, we calculate the mapping class group action explicitly. This completes the description of bordered Heegaard Floer homology for knot complements in terms of the knot Floer homology.

Heegaard Floer homology as morphism spaces

In this paper we prove another pairing theorem for bordered Floer homology. Unlike the original pairing theorem, this one is stated in terms of homomorphisms, not tensor products. The present formulation is closer in spirit to the usual TQFT framework, and allows a more direct comparison with Fukaya-categorical constructions. The result also leads to various dualities in bordered Floer homology.

Computing HF by factoring mapping classes

Bordered Heegaard Floer homology is an invariant for three-manifolds with boundary. In particular, this invariant associates to a handle decomposition of a surface F a differential graded algebra, and to an arc slide between two handle decompositions, a bimodule over the two algebras. In this paper, we describe these bimodules for arc slides explicitly, and then use them to give a combinatorial description of HF^ of a closed three-manifold, as well as the bordered Floer homology of any 3-manifold with boundary.

A Steenrod square on Khovanov homology

In a previous paper, we defined a space-level version X(L) of Khovanov homology. This induces an action of the Steenrod algebra on Khovanov homology. In this paper, we describe the first interesting operation, Sq^2:Kh^{i,j}(L) -> Kh^{i+2,j}(L). We compute this operation for all links up to 11 crossings; this, in turn, determines the stable homotopy type of X(L) for all such links.

Combinatorial cobordism maps in hat Heegaard Floer theory

In a previous article, Sarkar and Wang [15] gave a combinatorial description of the hat version of Heegaard Floer homology for three-manifolds. Given a cobordism between two connected three-manifolds, there is an induced map between their Heegaard Floer homologies. Assume that the first homology group of each boundary component surjects onto the first homology group of the cobordism (modulo torsion). Under this assumption, we present a procedure for finding the rank of the induced Heegaard Floer map combinatorially, in the hat version.

A faithful linear-categorical action of the mapping class group of a surface with boundary

We show that the action of the mapping class group on bordered Floer homology in the second to extremal spin^c-structure is faithful. This paper is designed partly as an introduction to the subject, and much of it should be readable without a background in Floer homology.

Tour of bordered Floer theory

Heegaard Floer theory is a kind of topological quantum field theory (TQFT), assigning graded groups to closed, connected, oriented 3-manifolds and group homomorphisms to smooth, oriented four-dimensional cobordisms. Bordered Heegaard Floer homology is an extension of Heegaard Floer homology to 3-manifolds with boundary, with extended-TQFT-type gluing properties. In this survey, we explain the formal structure and construction of bordered Floer homology and sketch how it can be used to compute some aspects of Heegaard Floer theory.

On transverse invariants from Khovanov homology

O. Plamenevskaya associated to each transverse knot K an element of the Khovanov homology of K. In this paper, we give two refinements of Plamenevskayas invariant, one valued in Bar-Natans deformation of the Khovanov complex and another as a cohomotopy element of the Khovanov spectrum. We show that the first of these refinements is invariant under negative flypes and SZ moves; this implies that Plamenevskayas class is also invariant under these moves. We go on to show that for small-crossing transverse knots K, both refinements are determined by the classical invariants of K.