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We prove an exact sequence for –twisted Heegaard Floer homology. As a corollary, given a torus bundle over the circle and a cohomology class which evaluates nontrivially on the fiber, we compute the Heegaard Floer homology of with twisted coefficients in the universal Novikov ring.
We show that certain classes of contact –manifolds do not admit nonseparating contact type embeddings into any closed symplectic –manifold, eg this is the case for all contact manifolds that are (partially) planar or have Giroux torsion. The latter implies that manifolds with Giroux torsion do not admit contact type embeddings into any closed symplectic –manifold. Similarly, there are symplectic –manifolds that can admit smoothly embedded nonseparating hypersurfaces, but not of contact type: we observe that this is the case for all symplectic ruled surfaces.
We define a set of “second-order” –signature invariants for any algebraically slice knot. These obstruct a knot’s being a slice knot and generalize Casson–Gordon invariants, which we consider to be “first-order signatures”. As one application we prove: If is a genus one slice knot then, on any genus one Seifert surface , there exists a homologically essential simple closed curve of self-linking zero, which has vanishing zero-th order signature and a vanishing first-order signature. This extends theorems of Cooper and Gilmer. We introduce a geometric notion, that of a derivative of a knot with respect to a metabolizer. We also introduce a new relation, generalizing homology cobordism, called null-bordism.
In this paper, we give a structure theorem for c-incompressible Conway spheres in link complements in terms of the standard height function on . We go on to define the generalized Conway product of two links and . Provided satisfies minor additional hypotheses, we prove the lower bound for the bridge number of the generalized Conway product where is the distinguished factor. Finally, we present examples illustrating that this lower bound is tight.
We study the behavior of the Ozsváth–Szabó and Rasmussen knot concordance invariants and on , the –cable of a knot where and are relatively prime. We show that for every knot and for any fixed positive integer , both of the invariants evaluated on differ from their value on the torus knot by fixed constants for all but finitely many . Combining this result together with Hedden’s extensive work on the behavior of on –cables yields bounds on the value of on any –cable of . In addition, several of Hedden’s obstructions for cables bounding complex curves are extended.
Consider the kernel of the Magnus representation of the Torelli group and the kernel of the Burau representation of the braid group. We prove that for and for the groups and have infinite rank first homology. As a consequence we conclude that neither group has any finite generating set. The method of proof in each case consists of producing a kind of “Johnson-type” homomorphism to an infinite rank abelian group, and proving the image has infinite rank. For the case of , we do this with the assistance of a computer calculation.
For any positive integer , we give a 2–component surface-link such that is orientable, is non-orientable and the triple point number of is equal to . To give lower bounds of the triple point numbers, we use symmetric quandle cocycle invariants.
We prove that the class of topological knot types that are both Legendrian simple and satisfy the uniform thickness property (UTP) is closed under cabling. An immediate application is that all iterated cabling knot types that begin with negative torus knots are Legendrian simple. We also examine, for arbitrary numbers of iterations, iterated cablings that begin with positive torus knots, and establish the Legendrian simplicity of large classes of these knot types, many of which also satisfy the UTP. In so doing we obtain new necessary conditions for both the failure of the UTP and Legendrian nonsimplicity in the class of iterated torus knots, including specific conditions on knot types.
Let a torus act effectively on a compact connected cooriented contact manifold, and let be the natural momentum map on the symplectization. We prove that, if is bigger than 2, the union of the origin with the image of is a convex polyhedral cone, the nonzero level sets of are connected (while the zero level set can be disconnected), and the momentum map is open as a map to its image. This answers a question posed by Eugene Lerman, who proved similar results when the zero level set is empty. We also analyze examples with .
Let , let be an orientable complete finite-volume hyperbolic –manifold with compact (possibly empty) geodesic boundary, and let and be the Riemannian volume and the simplicial volume of . A celebrated result by Gromov and Thurston states that if then , where is the volume of the regular ideal geodesic –simplex in hyperbolic –space. On the contrary, Jungreis and Kuessner proved that if then .
We prove here that for every there exists (only depending on and ) such that if , then . As a consequence we show that for every there exists a compact orientable hyperbolic –manifold with nonempty geodesic boundary such that .
Our argument also works in the case of empty boundary, thus providing a somewhat new proof of the proportionality principle for noncompact finite-volume hyperbolic –manifolds without geodesic boundary.
We state the generating hypothesis in the homotopy category of –spectra for a compact Lie group and prove that if is finite, then the generating hypothesis implies the strong generating hypothesis, just as in the non-equivariant case. We also give an explicit counterexample to the generating hypothesis in the category of rational –equivariant spectra.
Given a knot and an representation of its group that is conjugate to its dual, the representation that replaces each matrix with its inverse-transpose, the associated twisted Reidemeister torsion is reciprocal. An example is given of a knot group and representation for which the twisted Reidemeister torsion is not reciprocal.
A bottom tangle is a tangle in a cube consisting of arc components whose boundary points are placed on the bottom, and every link can be represented as the closure of a bottom tangle. The universal invariant of –component bottom tangles takes values in the –fold completed tensor power of the quantized enveloping algebra , and has a universality property for the colored Jones polynomials of –component links via quantum traces in finite dimensional representations. In the present paper, we prove that if the closure of a bottom tangle is a ribbon link, then the universal invariant of is contained in a certain small subalgebra of the completed tensor power of . As an application, we prove that ribbon links have stronger divisibility by cyclotomic polynomials than algebraically split links for Habiro’s reduced version of the colored Jones polynomials.
We study topology of configuration spaces of planar linkages having one leg of variable length. Such telescopic legs are common in modern robotics where they are used for shock absorbtion and serve a variety of other purposes. Using a Morse theoretic technique, we compute explicitly, in terms of the metric data, the Betti numbers of configuration spaces of these mechanisms.
Let be an odd prime, and fix integers and such that . We give a –local homotopy decomposition for the loop space of the complex Stiefel manifold . Similar decompositions are given for the loop space of the real and symplectic Stiefel manifolds. As an application of these decompositions, we compute upper bounds for the –exponent of . Upper bounds for –exponents in the stable range and are computed as well.
We extend Matveev’s complexity of –manifolds to PL compact manifolds of arbitrary dimension, and we study its properties. The complexity of a manifold is the minimum number of vertices in a simple spine. We study how this quantity changes under the most common topological operations (handle additions, finite coverings, drilling and surgery of spheres, products, connected sums) and its relations with some geometric invariants (Gromov norm, spherical volume, volume entropy, systolic constant).
Complexity distinguishes some homotopically equivalent manifolds and is positive on all closed aspherical manifolds (in particular, on manifolds with nonpositive sectional curvature). There are finitely many closed hyperbolic manifolds of any given complexity. On the other hand, there are many closed –manifolds of complexity zero (manifolds without –handles, doubles of –handlebodies, infinitely many exotic K3 surfaces, symplectic manifolds with arbitrary fundamental group).
In this paper, we study the growth with respect to dimension of quite general homotopy invariants applied to the CW skeleta of spaces. This leads to upper estimates analogous to the classical “dimension divided by connectivity” bound for Lusternik–Schnirelmann category. Our estimates apply, in particular, to the Clapp–Puppe theory of –category. We use (which is –category with the collection of –dimensional CW complexes), to reinterpret in homotopy-theoretical terms some recent work of Dranishnikov on the Lusternik–Schnirelmann category of spaces with fundamental groups of finite cohomological dimension. Our main result is the inequality , which implies and strengthens the main theorem of Dranishnikov [Algebr. Geom. Topol. 10 (2010) 917–924].
We describe a new –spectrum for connective –theory formed from spaces of operators which have certain nice spectral properties, and which fulfill a connectivity condition. The spectral data of such operators can equivalently be described by certain Clifford-linear, symmetric configurations on the real axis; in this sense, our model for stands between an older one by Segal, who uses nonsymmetric configurations without Clifford-structure on spheres, and the well-known Atiyah–Singer model for using Clifford-linear Fredholm operators. Dropping the connectivity condition we obtain operator spaces . These are homotopy equivalent to the spaces of –dimensional supersymmetric Euclidean field theories of degree which were defined by Stolz and Teichner (in terms of certain homomorphisms of super semigroups). They showed that the are homotopy equivalent to and gave the idea for the connection between and . We can derive a homotopy equivalent version of the –spectrum in terms of “field theory type” super semigroup homomorphisms. Tracing back our connectivity condition to the functorial language of field theories provides a candidate for connective –dimensional Euclidean field theories, , and might result in a more general criterion for instance for a connective version of –dimensional such theories (which are conjectured to yield a spectrum for ).
In this article we study a partial ordering on knots in where if there is an epimorphism from the knot group of onto the knot group of which preserves peripheral structure. If is a –bridge knot and , then it is known that must also be –bridge. Furthermore, Ohtsuki, Riley and Sakuma give a construction which, for a given –bridge knot , produces infinitely many –bridge knots with . After characterizing all –bridge knots with or less distinct boundary slopes, we use this to prove that in any such pair, is either a torus knot or has 5 or more distinct boundary slopes. We also prove that –bridge knots with exactly 3 distinct boundary slopes are minimal with respect to the partial ordering. This result provides some evidence for the conjecture that all pairs of –bridge knots with arise from the Ohtsuki–Riley–Sakuma construction.