A spanning tree cohomology theory for links

In their recent preprint, Baldwin, Ozsváth and Szabó defined a twisted version (with coefficients in a Novikov ring) of a spectral sequence, previously defined by Ozsváth and Szabó, from Khovanov homology to Heegaard–Floer homology of the branched double cover along a link. In their preprint, they give a combinatorial interpretation of the _E3$E_3$-term of their spectral sequence. The main purpose of the present paper is to prove directly that this _E3$E_3$-term is a link invariant. We also give some concrete examples of computation of the invariant.     

Rational homology spheres and the four-ball genus of knots

Using the Heegaard Floer homology of Ozsváth and Szabó we investigate obstructions to a rational homology sphere bounding a four-manifold with a definite intersection pairing. As an application we obtain new lower bounds for the four-ball genus of Montesinos links.     

Seiberg–Witten and Casson–Walker Invariants for Rational Homology 3-Spheres

We consider a modified version of the Seiberg–Witten invariants for rational homology 3-spheres, obtained by adding to the original invariants a correction term which is a combination of -invariants. We show that these modified invariants are topological invariants. We prove that an averaged version of these modified invariants equals the Casson–Walker invariant. In particular, this result proves an averaged version of a conjecture of Ozsváth and Szabó on the equivalence between their invariant and the Seiberg–Witten invariant of rational homology 3-spheres.     

An Ozsváth–Szabó Floer homology invariant of knots in a contact manifold

Using the knot Floer homology filtration, we define invariants associated to a knot in a three-manifold possessing non-vanishing Floer co(homology) classes. In the case of the Ozsváth–Szabó contact invariant we obtain an invariant of knots in a contact three-manifold. This invariant provides an upper bound for the Thurston–Bennequin plus rotation number of any Legendrian realization of the knot. We use it to demonstrate the first systematic construction of prime knots in contact manifolds other than S³ with negative maximal Thurston–Bennequin invariant. Perhaps more interesting, our invariant provides a criterion for an open book to induce a tight contact structure. A corollary is that if a manifold possesses contact structures with distinct non-vanishing Ozsváth–Szabó invariants, then any fibered knot can realize the classical Eliashberg–Bennequin bound in at most one of these contact structures.     

Quadratic functions and complex spin structures on three-manifolds

We show how the space of complex spin structures of a closed oriented three-manifold embeds naturally into a space of quadratic functions associated to its linking pairing. Besides, we extend the Goussarov-Habiro theory of finite type invariants to the realm of compact oriented three-manifolds equipped with a complex spin structure. Our main result states that two closed oriented three-manifolds endowed with a complex spin structure are undistinguishable by complex spin invariants of degree zero if, and only if, their associated quadratic functions are isomorphic.     

Donaldson's theorem,Heegaard Floer homology,and knots with unknotting number one

We establish an obstruction to unknotting an alternating knot by a single crossing change. The obstruction is lattice-theoretic in nature, and combines Donaldson's diagonalization theorem with an obstruction developed by Ozsváth and Szabó using Heegaard Floer homology. As an application, we enumerate the alternating 3-braid knots with unknotting number one, and show that each has an unknotting crossing in its standard alternating diagram.     

Holomorphic triangles and invariants for smooth four-manifolds

In earlier articles, the authors introduced invariants for closed, oriented three-manifolds, defined using a variant of Lagrangian Floer homology in the symmetric products of Riemann surfaces. The aim of this article is to introduce invariants of oriented, smooth four-manifolds, built using these Floer homology groups. This four-dimensional theory also endows the corresponding three-dimensional theories with additional structure: an absolute grading of certain of its Floer homology groups.     

Moduli for pointed convex domains

Summary A moduli space for the class of pointed strictly linearly convex domains in ⁿ is obtained. It is shown that the space of pointed smoothly bounded strictly linearly convex domains with a fixed indicatrix is parameterized by a class of deformations of the CR structure of the boundary of the indicatrix. These deformations are constructed by using the circular representation of a domain to pull back its complex structure tensor to the indicatrix. A careful study of the pull back structure shows that the allowable deformations are parameterized by a class of complex Hamiltonian vector fields. The proof of this fact is based on the Folland-Stein estimates for the complex of the boundary of the indicatrix.The paper is related to one of László Lempert, Holomorphic invariants, normal forms and moduli space of convex domains. Ann. Math128, 47–78 (1988), where other modular data for pointed convex domains were constructed. A method of recovering Lempert's modular data from the deformation moduli is given.Oblatum 26-IX-1989 & 22-III-1990Partially supported by an NSERC grant.The second author wishes to thank the University of Toronto and the Mathematical Sciences Research Institute at Berkeley, where portions of the paper were written.     

On three-manifolds dominated by circle bundles

We determine which three-manifolds are dominated by products. The result is that a closed, oriented, connected three-manifold is dominated by a product if and only if it is finitely covered either by a product or by a connected sum of copies of S ² × S ¹. This characterization can also be formulated in terms of Thurston geometries, or in terms of purely algebraic properties of the fundamental group. We also determine which three-manifolds are dominated by non-trivial circle bundles, and which three-manifold groups are presentable by products.     

Vassiliev invariants of Legendrian, transverse, and framed knots in contact three-manifolds

We show that for a large class of contact three-manifolds the groups of Vassiliev invariants of Legendrian and of framed knots are canonically isomorphic. As a corollary, we obtain that the group of finite order Arnold's J⁺-type invariants of wave fronts on a surface F is isomorphic to the group of Vassiliev invariants of framed knots in the spherical cotangent bundle ST∗F of F.On the other hand, we construct the first examples of contact manifolds for which Vassiliev invariants of Legendrian knots can distinguish Legendrian knots that realize isotopic framed knots and are homotopic as Legendrian immersions.     

Taut contact circles and bi-contact metric structures on three-manifolds

Geiges and Gonzalo (Invent. Math. 121:147–209 1995, J. Differ. Geom. 46:236–286 1997, Acta. Math. Vietnam 38:145–164 2013) introduced and studied the notion of taut contact circle on a three-manifold. In this paper, we introduce a Riemannian approach to the study of taut contact circles on three-manifolds. We characterize the existence of a taut contact metric circle and of a bi-contact metric structure. Then, we give a complete classification of simply connected three-manifolds which admit a bi-H-contact metric structure. In particular, a simply connected three-manifold admits a homogeneous bi-contact metric structure if and only if it is diffeomorphic to one of the following Lie groups: SU(2), \({\widetilde{SL}}(2,{\mathbb {R}})\), \({\widetilde{E}}(2)\), E(1, 1). Moreover, we obtain a classification of three-manifolds which admit a Cartan structure \((\eta _1,\eta _2)\) with the so-called Webster function \({\mathcal {W}}\) constant along the flow of \(\xi _1\) (equivalently \(\xi _2\)). Finally, we study the metric cone, i.e., the symplectization, of a bi-contact metric three-manifold. In particular, the notion of bi-contact metric structure is related to the notions of conformal symplectic couple (in the sense of Geiges (Duke Math. J. 85:701–711 1996)) and symplectic pair (in the sense of Bande and Kotschick (Trans. Am. Math. Soc. 358(4):1643–1655 2005)).     

Nonzero degree maps between closed orientable three-manifolds

This paper adresses the following problem: Given a closed orientable three-manifold , are there at most finitely many closed orientable three-manifolds 1-dominated by ? We solve this question for the class of closed orientable graph manifolds. More precisely the main result of this paper asserts that any closed orientable graph manifold 1-dominates at most finitely many orientable closed three-manifolds satisfying the Poincaré-Thurston Geometrization Conjecture. To prove this result we state a more general theorem for Haken manifolds which says that any closed orientable three-manifold 1-dominates at most finitely many Haken manifolds whose Gromov simplicial volume is sufficiently close to that of .

     

Invariants of Spin Three-Manifolds From Chern-Simons Theory and Finite-Dimensional Hopf Algebras

A version of Kirby calculus for spin and framed three-manifolds is given and is used to construct invariants of spin and framed three-manifolds in two situations. The first is ribbon *-categories which possess odd degenerate objects. This case includes the quantum group situations corresponding to the half-integer level Chern-Simons theories conjectured to give spin TQFTs by Dijkgraaf and Witten (1990, Commun. Math. Phys.129, 393-429). In particular, the spin invariants constructed by Kirby and Melvin (1991, Invent. Math.105, 473-545) are shown to be identical to the invariants associated to SO(3). Second, an invariant of spin manifolds analogous to the Hennings invariant is constructed beginning with an arbitrary factorizable, unimodular quasitriangular Hopf algebra. In particular a framed manifold invariant is associated to every finite-dimensional Hopf algebra via its quantum double, and is conjectured to be identical to Kuperberg's noninvolutory invariant of framed manifolds associated to that Hopf algebra.     

Controlled topology invariants of translation actions

We develop invariants Ωn of a translation action of a group on Rm analogous to the Bieri-Neumann-Strebel-Renz invariants Σn. The invariants Σn were defined to be the set of “directions” e∈_∂∞Rm such that a suitable universal G-space is (n−1)-connected over the half-spaces defined by e. We replace half-spaces by topologically more natural neighborhoods of e to obtain the new invariants Ωn. The invariants Σn and Ωn are related as follows: e∈Ωn if and only if every e^′ in an open -neighborhood of e lies in Σn.     

Serre-Taubes duality for pseudoholomorphic curves

According to Taubes, the Gromov invariants of a symplectic four-manifold X with b₊>1 satisfy the duality Gr(α)=±Gr(κ−α), where κ is Poincaré dual to the canonical class. Extending joint work with Simon Donaldson, we interpret this result in terms of Serre duality on the fibres of a Lefschetz pencil on X, by proving an analogous symmetry for invariants counting sections of associated bundles of symmetric products. Using similar methods, we give a new proof of an existence theorem for symplectic surfaces in four-manifolds with b₊=1 and b₁=0. This reproves another theorem due to Taubes: two symplectic homology projective planes with negative canonical class and equal volume are symplectomorphic.