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Most of the years of study of the set of knot concordance classes, , has focused on its structure as an abelian group. Here we take a different approach, namely we study as a metric space admitting many natural geometric operators. We focus especially on the coarse geometry of satellite operators. We consider several knot concordance spaces, corresponding to different categories of concordance, and two different metrics. We establish the existence of quasi-–flats for every , implying that admits no quasi-isometric embedding into a finite product of (Gromov) hyperbolic spaces. We show that every satellite operator is a quasihomomorphism . We show that winding number one satellite operators induce quasi-isometries with respect to the metric induced by slice genus. We prove that strong winding number one patterns induce isometric embeddings for certain metrics. By contrast, winding number zero satellite operators are bounded functions and hence quasicontractions. These results contribute to the suggestion that is a fractal space. We establish various other results about the large-scale geometry of arbitrary satellite operators.
We give a description of the factorization homology and topological Hochschild cohomology of Thom spectra arising from –fold loop maps , where is an –fold loop space. We describe the factorization homology as the Thom spectrum associated to a certain map , where is the factorization homology of with coefficients in . When is framed and is –connected, this spectrum is equivalent to a Thom spectrum of a virtual bundle over the mapping space ; in general, this is a Thom spectrum of a virtual bundle over a certain section space. This can be viewed as a twisted form of the nonabelian Poincaré duality theorem of Segal, Salvatore and Lurie, which occurs when is nullhomotopic. This result also generalizes the results of Blumberg, Cohen and Schlichtkrull on the topological Hochschild homology of Thom spectra, and of Schlichtkrull on higher topological Hochschild homology of Thom spectra. We use this description of the factorization homology of Thom spectra to calculate the factorization homology of the classical cobordism spectra, spectra arising from systems of groups and the Eilenberg–Mac Lane spectra , and . We build upon the description of the factorization homology of Thom spectra to study the ( and higher) topological Hochschild cohomology of Thom spectra, which enables calculations and a description in terms of sections of a parametrized spectrum. If is a closed manifold, Atiyah duality for parametrized spectra allows us to deduce a duality between topological Hochschild homology and topological Hochschild cohomology, recovering string topology operations when is nullhomotopic. In conjunction with the higher Deligne conjecture, this gives –structures on a certain family of Thom spectra, which were not previously known to be ring spectra.
Given a filtration of a commutative monoid in a symmetric monoidal stable model category , we construct a spectral sequence analogous to the May spectral sequence whose input is the higher order topological Hochschild homology of the associated graded commutative monoid of , and whose output is the higher order topological Hochschild homology of . We then construct examples of such filtrations and derive some consequences: for example, given a connective commutative graded ring , we get an upper bound on the size of the –groups of –ring spectra such that .
The Gauss–Bonnet formula for classical translation surfaces relates the cone angle of the singularities (geometry) to the genus of the surface (topology). When considering more general translation surfaces, we observe so-called wild singularities for which the notion of cone angle is not applicable any more.
We study whether there still exist relations between the geometry and the topology for translation surfaces with wild singularities. By considering short saddle connections, we determine under which conditions the existence of a wild singularity implies infinite genus. We apply this to show that parabolic or essentially finite translation surfaces with wild singularities have infinite genus.
We prove a modulo invariance for the number of Klein-bottle leaves in taut foliations. Given two smooth cooriented taut foliations, assume that every Klein-bottle leaf has nontrivial linear holonomy, and assume that the two foliations can be smoothly deformed to each other through taut foliations. We prove that the numbers of Klein-bottle leaves in these two foliations must have the same parity.
The work of Volker Puppe and Matthias Kreck exhibited some intriguing connections between the algebraic topology of involutions on closed manifolds and the combinatorics of self-dual binary codes. On the other hand, the work of Michael Davis and Tadeusz Januszkiewicz brought forth a topological analogue of smooth, real toric varieties, known as “small covers”, which are closed smooth manifolds equipped with some actions of elementary abelian –groups whose orbit spaces are simple convex polytopes. Building on these works, we find various new connections between all these topological and combinatorial objects and obtain some new applications to the study of self-dual binary codes, as well as colorability of polytopes. We first show that a small cover over a simple –polytope produces a self-dual code in the sense of Kreck and Puppe if and only if is –colorable and is odd. Then we show how to describe such a self-dual binary code in terms of the combinatorics of . Moreover, we can construct a family of binary codes , for , from an arbitrary simple –polytope . Then we give some necessary and sufficient conditions for to be self-dual. A spinoff of our study of such binary codes gives some new ways to judge whether a simple –polytope is –colorable in terms of the associated binary codes . In addition, we prove that the minimum distance of the self-dual binary code obtained from a –colorable simple –polytope is always .
We prove a homological stability theorem for the diffeomorphism groups of high-dimensional manifolds with boundary, with respect to forming the boundary connected sum with the product for . In a recent joint paper with B Botvinnik, we prove that there is an isomorphism
in the case that . By combining this “stable homology” calculation with the homological stability theorem of this paper, we obtain the isomorphism
Framed flow categories were introduced by Cohen, Jones and Segal as a way of encoding the flow data associated to a Floer functional. A framed flow category gives rise to a CW complex with one cell for each object of the category. The idea is that the Floer invariant should take the form of the stable homotopy type of the resulting complex, recovering the Floer cohomology as its singular cohomology. Such a framed flow category was produced, for example, by Lipshitz and Sarkar from the input of a knot diagram, resulting in a stable homotopy type generalising Khovanov cohomology.
We give moves that change a framed flow category without changing the associated stable homotopy type. These are inspired by moves that can be performed in the Morse–Smale case without altering the underlying smooth manifold. We posit that if two framed flow categories represent the same stable homotopy type then a finite sequence of these moves is sufficient to connect the two categories. This is directed towards the goal of reducing the study of framed flow categories to a combinatorial calculus.
We provide examples of calculations performed with these moves (related to the Khovanov framed flow category), and prove some general results about the simplification of framed flow categories via these moves.
We prove affirmatively the conjecture raised by J Mostovoy (Topology 41 (2002) 435–450); the space of short ropes is weakly homotopy equivalent to the classifying space of the topological monoid (or category) of long knots in . We make use of techniques developed by S Galatius and O Randal-Williams (Geom. Topol. 14 (2010) 1243–1302) to construct a manifold space model of the classifying space of the space of long knots, and we give an explicit map from the space of short ropes to the model in a geometric way.
Let for be the special linear group and be a closed aspherical manifold. It is proved that when , a group action of on by homeomorphisms is trivial if and only if the induced group homomorphism is trivial. For (almost) flat manifolds, we prove a similar result in terms of holonomy groups. In particular, when is nilpotent, the group cannot act nontrivially on when . This confirms a conjecture related to Zimmer’s program for these manifolds.
Kakimizu complexes of Seifert fibered spaces can be described as either horizontal or vertical, depending on what type of surfaces represent their vertices. Horizontal Kakimizu complexes are shown to be trivial. Each vertical Kakimizu complex is shown to be isomorphic to a Kakimizu complex of the base orbifold minus its singular points.
We prove a conjecture of Blumberg and Hill regarding the existence of –operads associated to given sequences of families of subgroups of . For every such sequence, we construct a model structure on the category of –operads, and we use these model structures to define –operads, generalizing the notion of an –operad, and to prove the Blumberg–Hill conjecture. We then explore questions of admissibility, rectification, and preservation under left Bousfield localization for these –operads, obtaining some new results as well for –operads.
We discuss what it means for a symmetric monoidal category to be a module over a commutative semiring category. Each of the categories of (1) cartesian monoidal categories, (2) semiadditive categories, and (3) connective spectra can be recovered in this way as categories of modules over a commutative semiring category (or –category in the last case). This language provides a simultaneous generalization of the formalism of algebraic theories (operads, PROPs, Lawvere theories) and stable homotopy theory, with essentially a variant of algebraic K–theory bridging between the two.
We study several properties of the completed group ring and the completed Alexander modules of knots. Then we prove that if the profinite completions of the groups of two knots and are isomorphic, then their Alexander polynomials and coincide.
We calculate the fundamental group of locally standard –manifolds under the assumption that the principal –bundle obtained from the free –orbits is trivial. This family of manifolds contains nonsingular toric varieties which may be noncompact, quasitoric manifolds and toric origami manifolds with coörientable folding hypersurface. Although the fundamental groups of the above three kinds of manifolds are well-studied, we give a uniform and simple method to generalize the formulas of their fundamental groups.
For a continuous angle-valued map defined on a compact ANR , a field and any integer , one proposes a refinement of the Novikov–Betti numbers of the pair and a refinement of the Novikov homology of , where denotes the integral degree one cohomology class represented by . The refinement is a configuration of points, with multiplicity located in identified to , whose total cardinality is the Novikov–Betti number of the pair. The refinement is a configuration of submodules of the Novikov homology whose direct sum is isomorphic to the Novikov homology and with the same support as of . When , the configuration is convertible into a configuration of mutually orthogonal closed Hilbert submodules of the –homology of the infinite cyclic cover of defined by , which is an –Hilbert module. One discusses the properties of these configurations, namely robustness with respect to continuous perturbation of the angle-values map and the Poincaré duality and one derives some computational applications in topology. The main results parallel the results for the case of real-valued map but with Novikov homology and Novikov–Betti numbers replacing standard homology and standard Betti numbers.
We study the number of ends of a Schreier graph of a hyperbolic group. Let be a hyperbolic group and let be a subgroup of . In general, there is no algorithm to compute the number of ends of a Schreier graph of the pair . However, assuming that is a quasiconvex subgroup of , we construct an algorithm.
We use classical techniques to answer some questions raised by Daniele Celoria about almost-concordance of knots in arbitrary closed –manifolds. We first prove that, given , for any nontrivial element there are infinitely many distinct smooth almost-concordance classes in the free homotopy class of the unknot. In particular we consider these distinct smooth almost-concordance classes on the boundary of a Mazur manifold and we show none of these distinct classes bounds a PL–disk in the Mazur manifold, but all the representatives we construct are topologically slice. We also prove that all knots in the free homotopy class of in are smoothly concordant.