Dylan P. Thurston

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.

On combinatorial link Floer homology

Link Floer homology is an invariant for links defined using a suitable version of Lagrangian Floer homology. In an earlier paper, this invariant was given a combinatorial description with mod 2 coefficients. In the present paper, we give a self-contained presentation of the basic properties of link Floer homology, including an elementary proof of its invariance. We also fix signs for the differentials, so that the theory is defined with integer coefficients.

Cluster Algebras and Triangulated Surfaces Part II: Lambda Lengths

For any cluster algebra whose underlying combinatorial data can be encoded by a bordered surface with marked points, we construct a geometric realization in terms of suitable decorated Teichmueller space of the surface. On the geometric side, this requires opening the surface at each interior marked point into an additional geodesic boundary component. On the algebraic side, it relies on the notion of a non-normalized cluster algebra and the machinery of tropical lambda lengths. Our model allows for an arbitrary choice of coefficients which translates into a choice of a family of integral laminations on the surface. It provides an intrinsic interpretation of cluster variables as renormalized lambda lengths of arcs on the surface. Exchange relations are written in terms of the shear coordinates of the laminations, and are interpreted as generalized Ptolemy relations for lambda lengths. This approach gives alternative proofs for the main structural results from our previous paper, removing unnecessary assumptions on the surface.

Legendrian knots, transverse knots and combinatorial Floer homology

Using the combinatorial approach to knot Floer homology, we define an invariant for Legendrian knots in the three-sphere, which takes values in link Floer homology. This invariant can be used to also construct an invariant of transverse knots.

Wheels, wheeling, and the Kontsevich integral of the Unknot

We conjecture an exact formula for the Kontsevich integral of the unknot, and also conjecture a formula (also conjectured independently by Deligne [De]) for the relation between the two natural products on the space of uni-trivalent diagrams. The two formulas use the related notions of “Wheels” and “Wheeing”. We prove these formulas ‘on the level of Lie algebras’ using standard techniques from the theory of Vassiliev invariants and the theory of Lie algebras. In a brief epilogue we report on recent proofs of our full conjectures, by Kontsevich [Ko2] and by DBN, DPT, and T. Q. T. Le, [BLT].

The complex volume of

For a compact 3-manifold M with arbitrary (possibly empty) boundary, we give a parametrization of the set of conjugacy classes of boundary-unipotent representations of the fundamental group of M into SL(n,C). Our parametrization uses Ptolemy coordinates, which are inspired by coordinates on higher Teichmueller spaces due to Fock and Goncharov. We show that a boundary-unipotent representation determines an element in Neumanns extended Bloch group, and use this to obtain an efficient formula for the Cheeger-Chern-Simons invariant, and in particular for the volume. Computations for the census manifolds show that boundary-unipotent representations are abundant, and numerical comparisons with census volumes, suggest that the volume of a representation is an integral linear combination of volumes of hyperbolic 3-manifolds. This is in agreement with a conjecture of Walter Neumann, stating that the Bloch group is generated by hyperbolic manifolds.

\operatorname {SL}(n,\mathbb{C})

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.

-representations of 3-manifolds

We address the question: how common is it for a 3‐manifold to fiber over the circle? One motivation for considering this is to give insight into the fairly inscrutable Virtual Fibration Conjecture. For the special class of 3‐manifolds with tunnel number one, we provide compelling theoretical and experimental evidence that fibering is a very rare property. Indeed, in various precise senses it happens with probability 0. Our main theorem is that this is true for a measured lamination model of random tunnel number one 3‐manifolds. The first ingredient is an algorithm of K Brown which can decide if a given tunnel number one 3‐manifold fibers over the circle. Following the lead of Agol, Hass and W Thurston, we implement Brown’s algorithm very efficiently by working in the context of train tracks/interval exchanges. To analyze the resulting algorithm, we generalize work of Kerckhoff to understand the dynamics of splitting sequences of complete genus 2 interval exchanges. Combining all of this with a “magic splitting sequence” and work of Mirzakhani proves the main theorem. The 3‐manifold situation contrasts markedly with random 2‐generator 1‐relator groups; in particular, we show that such groups “fiber” with probability strictly between 0 and 1.

Heegaard Floer homology as morphism spaces

It has been known since 1954 that every 3-manifold bounds a 4-manifold. Thus, for instance, every 3-manifold has a surgery diagram. There are several proofs of this fact, but little attention has been paid to the complexity of the 4-manifold produced. Given a 3-manifold M 3 of complexity n, we construct a 4-manifold bounded by M of complexity , where the ‘complexity’ of a piecewise-linear manifold is the minimum number of n-simplices in a triangulation.The proof goes through the notion of ‘shadow complexity’ of a 3-manifold M. A shadow of M is a well-behaved 2-dimensional spine of a 4-manifold bounded by M. We further prove that, for a manifold M satisfying the geometrization conjecture with Gromov norm G and shadow complexity S, we have , for suitable constants , . In particular, the manifolds with shadow complexity 0 are the graph manifolds.In addition, we give an bound for the complexity of a spin 4-manifold bounding a given spin 3-manifold. We also show that every stable map from a 3-manifold M with Gromov norm G to has at least crossing singularities, and if M is hyperbolic there is a map with at most crossing singularities.

A random tunnel number one 3–manifold does not fiber over the circle

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.