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For closed 3–manifolds, Heegaard Floer homology is related to the Thurston norm through results due to Ozsváth and Szabó, Ni, and Hedden. For example, given a closed 3–manifold , there is a bijection between vertices of the polytope carrying the group and the faces of the Thurston norm unit ball that correspond to fibrations of over the unit circle. Moreover, the Thurston norm unit ball of is dual to the polytope of .
We prove a similar bijection and duality result for a class of 3–manifolds with boundary called sutured manifolds. A sutured manifold is essentially a cobordism between two possibly disconnected surfaces with boundary and . We show that there is a bijection between vertices of the sutured Floer polytope carrying the group and equivalence classes of taut depth-one foliations that form the foliation cones of Cantwell and Conlon. Moreover, we show that a function defined by Juhász, which we call the geometric sutured function, is analogous to the Thurston norm in this context. In some cases, this function is an asymmetric norm and our duality result is that appropriate faces of this norm’s unit ball subtend the foliation cones.
An important step in our work is the following fact: a sutured manifold admits a fibration or a taut depth-one foliation whose sole compact leaves are exactly the connected components of and , if and only if, there is a surface decomposition of the sutured manifold resulting in a product manifold.
We consider –equivariant principal –bundles over proper –CW–complexes with a prescribed family of local representations. We construct and analyze their classifying spaces for locally compact, second countable topological groups and with finite covering dimensions, where is almost connected.
This paper contains some contributions to the study of classifying spaces for tricategories, with applications to the homotopy theory of monoidal categories, bicategories, braided monoidal categories and monoidal bicategories. Any small tricategory has various associated simplicial or pseudosimplicial objects and we explore the relationship between three of them: the pseudosimplicial bicategory (so-called Grothendieck nerve) of the tricategory, the simplicial bicategory termed its Segal nerve and the simplicial set called its Street geometric nerve. We prove that the geometric realizations of all of these ‘nerves of the tricategory’ are homotopy equivalent. By using Grothendieck nerves we state the precise form in which the process of taking classifying spaces transports tricategorical coherence to homotopy coherence. Segal nerves allow us to prove that, under natural requirements, the classifying space of a monoidal bicategory is, in a precise way, a loop space. With the use of geometric nerves, we obtain simplicial sets whose simplices have a pleasing geometrical description in terms of the cells of the tricategory and we prove that, via the classifying space construction, bicategorical groups are a convenient algebraic model for connected homotopy –types.
How do Seifert surgeries on hyperbolic knots arise from those on torus knots? We approach this question from a networking viewpoint introduced by the authors in [Mem. Amer. Math. Soc. 217 (2012), no. 1021]. The Seifert surgery network is a –dimensional complex whose vertices correspond to Seifert surgeries; two vertices are connected by an edge if one Seifert surgery is obtained from the other by a single twist along a trivial knot called a seiferter or along an annulus cobounded by seiferters. Successive twists along a “hyperbolic seiferter” or a “hyperbolic annular pair” produce infinitely many Seifert surgeries on hyperbolic knots. In this paper, we investigate Seifert surgeries on torus knots that have hyperbolic seiferters or hyperbolic annular pairs, and obtain results suggesting that such surgeries are restricted.
We develop the properties of the sequential topological complexity , a homotopy invariant introduced by the third author as an extension of Farber’s topological model for studying the complexity of motion planning algorithms in robotics. We exhibit close connections of to the Lusternik–Schnirelmann category of cartesian powers of , to the cup length of the diagonal embedding , and to the ratio between homotopy dimension and connectivity of . We fully compute the numerical value of for products of spheres, closed –connected symplectic manifolds and quaternionic projective spaces. Our study includes two symmetrized versions of . The first one, unlike Farber and Grant’s symmetric topological complexity, turns out to be a homotopy invariant of ; the second one is closely tied to the homotopical properties of the configuration space of cardinality- subsets of . Special attention is given to the case of spheres.
For any hyperbolic genus-one –bridge knot in the –sphere, such as any hyperbolic twist knot, we show that the manifold resulting from –surgery on the knot has left-orderable fundamental group if the slope lies in some range, which depends on the knot.
Let be a nontrivial compression body without toroidal boundary components. Let be the –character variety of . We examine the dynamics of the action of on , and in particular, we find an open set, on which the action is properly discontinuous, that is strictly larger than the interior of the deformation space of marked hyperbolic –manifolds homotopy equivalent to .
An –move is a homotopy of wrinkled fibrations which deforms images of indefinite fold singularities like the Reidemeister move of type II. Variants of this move are contained in several important deformations of wrinkled fibrations. In this paper, we first investigate how monodromies are changed by this move. For a given fibration and its vanishing cycles, we then give an algorithm to obtain vanishing cycles in a single reference fiber of a fibration obtained by flip and slip, which is a sequence of homotopies increasing fiber genera. As an application of this algorithm, we give several examples of diagrams which were introduced by Williams to describe smooth –manifolds by a finite sequence of simple closed curves in a closed surface.
The classifying space of a topological group can be filtered by a sequence of subspaces , , using the descending central series of free groups. If is finite, describing them as homotopy colimits is convenient when applying homotopy theoretic methods. In this paper we introduce natural subspaces defined for a fixed prime . We show that is stably homotopy equivalent to a wedge of as runs over the primes dividing the order of . Colimits of abelian groups play an important role in understanding the homotopy type of these spaces. Extraspecial –groups are key examples, for which these colimits turn out to be finite. We prove that for extraspecial –groups of order , , does not have the homotopy type of a space, thus answering in a negative way a question posed by Adem. For a finite group , we compute the complex –theory of modulo torsion.
Given a link , we ask whether the components of bound disjoint, nullhomologous disks properly embedded in a simply connected positive-definite smooth –manifold; the knot case has been studied extensively by Cochran, Harvey and Horn. Such a –manifold is necessarily homeomorphic to a (punctured) . We characterize all links that are slice in a (punctured) in terms of ribbon moves and an operation which we call adding a generalized positive crossing. We find obstructions in the form of the Levine–Tristram signature function, the signs of the first author’s generalized Sato–Levine invariants, and certain Milnor’s invariants. We show that the signs of coefficients of the Conway polynomial obstruct a –component link from being slice in a single punctured and conjecture these are obstructions in general. These results have applications to the question of when a –manifold bounds a –manifold whose intersection form is that of some . For example, we show that any homology –sphere is cobordant, via a smooth positive-definite manifold, to a connected sum of surgeries on knots in .
We construct a Chern–Simons gauge theory for dg Lie and –infinity algebras on any one-dimensional manifold and quantize this theory using the Batalin–Vilkovisky formalism and Costello’s renormalization techniques. Koszul duality and derived geometry allow us to encode topological quantum mechanics, a nonlinear sigma model of maps from a –manifold into a cotangent bundle , as such a Chern–Simons theory. Our main result is that the effective action of this theory is naturally identified with the class of . From the perspective of derived geometry, our quantization constructs a projective volume form on the derived loop space that can be identified with the class.
A –dimensional open book determines a closed, oriented –manifold and a contact structure on . The contact structure is Stein fillable if is positive, ie can be written as a product of right-handed Dehn twists. Work of Wendl implies that when has genus zero the converse holds, that is
On the other hand, results by Wand [Phd thesis (2010)] and by Baker, Etnyre and Van Horn–Morris [J. Differential Geom. 90 (2012) 1-80] imply the existence of counterexamples to the above implication with of arbitrary genus strictly greater than one. The main purpose of this paper is to prove the implication holds under the assumption that is a one-holed torus and is a Heegaard Floer –space.
We give a geometric approach to an algorithm for deciding whether two hyperbolic –manifolds are homeomorphic. We also give an algebraic approach to the homeomorphism problem for geometric, but nonhyperbolic, –manifolds.
Suppose that is a homomorphism from the mapping class group of a nonorientable surface of genus with boundary components to . We prove that if , and , then factors through the abelianization of , which is for and for . If , and , then either has finite image (of order at most two if ), or it is conjugate to one of four “homological representations”. As an application we prove that for and , every homomorphism factors through the abelianization of .
We compute –invariants of certain nonuniform lattices in semisimple Lie groups by means of the Borel–Serre compactification of arithmetically defined locally symmetric spaces. The main results give new estimates for Novikov–Shubin numbers and vanishing –torsion for lattices in groups with even deficiency. We discuss applications to Gromov’s zero-in-the-spectrum conjecture as well as to a proportionality conjecture for the –torsion of measure-equivalent groups.