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We give an alternative construction of the Kontsevich–Kuperberg–Thurston invariant for rational homology –spheres. This construction is a generalization of the original construction of the Kontsevich–Kuperberg–Thurston invariant. As an application, we give a Morse homotopy theoretic description of the Kontsevich–Kuperberg–Thurston invariant (close to a description by Watanabe).
We perform two explicit computations of bordered Heegaard Floer invariants. The first is the type trimodule associated to the trivial –bundle over the pair of pants . The second is a bimodule that is necessary for self-gluing when two torus boundary components of a bordered manifold are glued to each other. Using the results of these two computations, we describe an algorithm for computing of any graph manifold.
We study a class of –manifolds called strong L–spaces, which by definition admit a certain type of Heegaard diagram that is particularly simple from the perspective of Heegaard Floer homology. We provide evidence for the possibility that every strong L–space is the branched double cover of an alternating link in the three-sphere. For example, we establish this fact for a strong L–space admitting a strong Heegaard diagram of genus 2 via an explicit classification. We also show that there exist finitely many strong L–spaces with bounded order of first homology; for instance, through order eight, they are connected sums of lens spaces. The methods are topological and graph-theoretic. We discuss many related results and questions.
Checkerboard surfaces in alternating link complements are used frequently to determine information about the link. However, when many crossings are added to a single twist region of a link diagram, the geometry of the link complement stabilizes (approaches a geometric limit), but a corresponding checkerboard surface increases in complexity with crossing number. In this paper, we generalize checkerboard surfaces to certain immersed surfaces, called twisted checkerboard surfaces, whose geometry better reflects that of the alternating link in many cases. We describe the surfaces, show that they are essential in the complement of an alternating link, and discuss their properties, including an analysis of homotopy classes of arcs on the surfaces in the link complement.
A relative category is a category with a chosen class of weak equivalences. Barwick and Kan produced a model structure on the category of all relative categories, which is Quillen equivalent to the Joyal model structure on simplicial sets and the Rezk model structure on simplicial spaces. We will prove that the underlying relative category of a model category or even a fibration category is fibrant in the Barwick–Kan model structure.
Weaving knots are alternating knots with the same projection as torus knots, and were conjectured by X-S Lin to be among the maximum volume knots for fixed crossing number. We provide the first asymptotically sharp volume bounds for weaving knots, and we prove that the infinite square weave is their geometric limit.
Every knot or link can be put in a bridge position with respect to a –sphere for some bridge number , where is the bridge number for . Such –bridge positions determine –plat projections for the knot. We show that if and the underlying braid of the plat has rows of twists and all the twisting coefficients have absolute values greater than or equal to three then the distance of the bridge sphere is exactly , where is the smallest integer greater than or equal to . As a corollary, we conclude that if such a diagram has rows then the bridge sphere defining the plat projection is the unique, up to isotopy, minimal bridge sphere for the knot or link. This is a crucial step towards proving a canonical (thus a classifying) form for knots that are “highly twisted” in the sense we define.
Handel and Mosher define the axis bundle for a fully irreducible outer automorphism in [Mem. Amer. Math. Soc. 1004 (2011)]. We give a necessary and sufficient condition for the axis bundle to consist of a unique periodic fold line. As a consequence, we give a setting, and means for identifying in this setting, when two elements of an outer automorphism group have conjugate powers.
Suppose is a knot in with bridge number and bridge distance greater than . We show that there are at most distinct minimal-genus Heegaard splittings of . These splittings can be divided into two families. Two splittings from the same family become equivalent after at most one stabilization. If has bridge distance at least , then two splittings from different families become equivalent only after stabilizations. Furthermore, we construct representatives of the isotopy classes of the minimal tunnel systems for corresponding to these Heegaard surfaces.
We prove that for any there exists a hyperbolic manifold such that and . This was conjectured by the authors in [Algebr. Geom. Topol. 13 (2013) 927–958, Conjecture 1.3].
The proof requires study of cosmetic surgery on links (equivalently, fillings of manifolds with boundary tori). There is no bound on the number of components of the link (or boundary components). For statements, see the second part of the introduction. Here are two examples of the results we obtain:
Let be a component of a link in . Then “most” slopes on cannot be completed to a cosmetic surgery on , unless becomes a component of a Hopf link.
Let be a manifold and . Then all but finitely many hyperbolic manifolds obtained by filling admit a geodesic shorter than . (Note that it is not true that there are only finitely many fillings fulfilling this condition.)
Donaldson [J. Differential Geom. 53 (1999) 205–236] showed that every closed symplectic –manifold can be given the structure of a topological Lefschetz pencil. Gay and Kirby [Geom. Topol. 20 (2016) 3097–3132] showed that every closed –manifold has a trisection. In this paper we relate these two structure theorems, showing how to construct a trisection directly from a topological Lefschetz pencil. This trisection is such that each of the three sectors is a regular neighborhood of a regular fiber of the pencil. This is a –dimensional analog of the following trivial –dimensional result: for every open book decomposition of a –manifold , there is a decomposition of into three handlebodies, each of which is a regular neighborhood of a page.
Given a limit sketch in which the cones have a finite connected base, we show that a model structure of “up to homotopy” models for this limit sketch in a suitable model category can be transferred to a Quillen-equivalent model structure on the category of strict models. As a corollary of our general result, we obtain a rigidification theorem which asserts in particular that any –space in the sense of Rezk is levelwise equivalent to one that satisfies the Segal conditions on the nose. There are similar results for dendroidal spaces and –fold Segal spaces.
For every H-space , the set of homotopy classes possesses a natural algebraic structure of a loop near-ring. Albeit one cannot say much about general loop near-rings, it turns out that those that arise from H-spaces are sufficiently close to rings to have a viable Krull–Schmidt type decomposition theory, which is then reflected into decomposition results of H-spaces. In the paper, we develop the algebraic theory of local loop near-rings and derive an algebraic characterization of indecomposable and strongly indecomposable H-spaces. As a consequence, we obtain unique decomposition theorems for products of H-spaces. In particular, we are able to treat certain infinite products of H-spaces, thanks to a recent breakthrough in the Krull–Schmidt theory for infinite products. Finally, we show that indecomposable finite –local H-spaces are automatically strongly indecomposable, which leads to an easy alternative proof of classical unique decomposition theorems of Wilkerson and Gray.
We prove the existence of minimal models for fibrations between dendroidal sets in the model structure for –operads, as well as in the covariant model structure for algebras and in the stable one for connective spectra. We also explain how our arguments can be used to extend the results of Cisinski (2014) and give the existence of minimal fibrations in model categories of presheaves over generalized Reedy categories of a rather common type. Besides some applications to the theory of algebras over –operads, we also prove a gluing result for parametrized connective spectra (or –spaces).
A subgroup of a group is commensurated in if for each , has finite index in both and . If there is a sequence of subgroups where is commensurated in for all , then is subcommensurated in . In this paper we introduce the notion of the simple connectivity at of a finitely generated group (in analogy with that for finitely presented groups). Our main result is this: if a finitely generated group contains an infinite finitely generated subcommensurated subgroup of infinite index in , then is one-ended and semistable at . If, additionally, is recursively presented and is finitely presented and one-ended, then is simply connected at . A normal subgroup of a group is commensurated, so this result is a strict generalization of a number of results, including the main theorems in works of G Conner and M Mihalik, B Jackson, V M Lew, M Mihalik, and J Profio. We also show that Grigorchuk’s group (a finitely generated infinite torsion group) and a finitely presented ascending HNN extension of this group are simply connected at , generalizing the main result of a paper of L Funar and D E Otera.
In any closed contact manifold of dimension at least , we construct examples of closed Legendrian submanifolds which are not diffeomorphic but whose Lagrangian cylinders in the symplectization are Hamiltonian isotopic.
Inspired by bordered Floer homology, we describe a type structure in Khovanov homology, which complements the type structure previously defined by the author. The type structure is a differential module over a certain algebra. This can be paired with the type structure to recover the Khovanov chain complex. The homotopy type of the type structure is a tangle invariant, and homotopy equivalences of the type structure result in chain homotopy equivalences on the Khovanov chain complex found from a pairing. We use this to simplify computations and introduce a modular approach to the computation of Khovanov homologies. Several examples are included, showing in particular how this approach computes the correct torsion summands for the Khovanov homology of connect sums. A lengthy appendix is devoted to establishing the theory of these structures over a characteristic-zero ring.