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Given an orientation-preserving self-diffeomorphism of a closed, orientable surface with genus at least two and an embedding of into a three-manifold , we construct a mutant manifold by cutting along and regluing by . We will consider whether there exist nontrivial gluings such that for any embedding, the manifold and its mutant have isomorphic Heegaard Floer homology. In particular, we will demonstrate that if is not isotopic to the identity map, then there exists an embedding of into a three-manifold such that the rank of the nontorsion summands of of differs from that of its mutant. We will also show that if the gluing map is isotopic to neither the identity nor the genus-two hyperelliptic involution, then there exists an embedding of into a three-manifold such that the total rank of of differs from that of its mutant.
We show that any integral second cohomology class of a closed manifold , , admits, as a Poincaré dual, a submanifold such that has a handle decomposition with no handles of index bigger than . In particular, if is an almost complex manifold of dimension at least , the complement can be given a structure of a Stein manifold.
The aim of this article is to establish the notion of bundle-type quasitoric manifolds and provide two classification results on them: (i) –bundle type quasitoric manifolds are weakly equivariantly homeomorphic if their cohomology rings are isomorphic, and (ii) quasitoric manifolds over are homeomorphic if their cohomology rings are isomorphic. In the latter case, there are only four quasitoric manifolds up to weakly equivariant homeomorphism which are not bundle-type.
We say that a given knot is detected by its knot Floer homology and –polynomial if whenever a knot has the same knot Floer homology and the same –polynomial as , then . In this paper we show that every torus knot is detected by its knot Floer homology and –polynomial. We also give a one-parameter family of infinitely many hyperbolic knots in each of which is detected by its knot Floer homology and –polynomial. In addition we give a cabling formula for the –polynomials of cabled knots in , which is of independent interest. In particular we give explicitly the –polynomials of iterated torus knots.
Ozsváth, Stipsicz and Szabó have defined a knot concordance invariant taking values in the group of piecewise linear functions on the closed interval . This paper presents a description of one approach to defining and proving its basic properties.
Let be an infinite discrete group and let be a classifying space for proper actions of . Every –equivariant vector bundle over gives rise to a compatible collection of representations of the finite subgroups of . We give the first examples of groups with a cocompact classifying space for proper actions admitting a compatible collection of representations of the finite subgroups of that does not come from a –equivariant (virtual) vector bundle over . This implies that the Atiyah–Hirzebruch spectral sequence computing the –equivariant topological –theory of has nonzero differentials. On the other hand, we show that for right-angled Coxeter groups this spectral sequence always collapses at the second page and compute the –theory of the classifying space of a right-angled Coxeter group.
We establish some facts about the behavior of the rational-geometric subvariety of the or character variety of a hyperbolic knot manifold under the restriction map to the or character variety of the boundary torus, and use the results to get some properties about the –polynomials and to prove the AJ conjecture for a certain class of knots in including in particular any –bridge knot over which the double branched cover of is a lens space of prime order.
Various models of –categories, including quasi-categories, complete Segal spaces, Segal categories, and naturally marked simplicial sets can be considered as the objects of an –cosmos. In a generic –cosmos, whose objects we call –categories, we introduce modules (also called profunctors or correspondences) between –categories, incarnated as spans of suitably defined fibrations with groupoidal fibers. As the name suggests, a module from to is an –category equipped with a left action of and a right action of , in a suitable sense. Applying the fibrational form of the Yoneda lemma, we develop a general calculus of modules, proving that they naturally assemble into a multicategory-like structure called a virtual equipment, which is known to be a robust setting in which to develop formal category theory. Using the calculus of modules, it is straightforward to define and study pointwise Kan extensions, which we relate, in the case of cartesian closed –cosmoi, to limits and colimits of diagrams valued in an –category, as introduced in previous work.
We develop a theory of chain complex double cobordism for chain complexes equipped with Poincaré duality. The resulting double cobordism groups are a refinement of the classical torsion algebraic –groups for localisations of a ring with involution. The refinement is analogous to the difference between metabolic and hyperbolic linking forms.
We apply the double –groups in high-dimensional knot theory to define an invariant for doubly slice –knots. We prove that the “stably doubly slice implies doubly slice” property holds (algebraically) for Blanchfield forms, Seifert forms and for the Blanchfield complexes of –knots for .
We show that the equation associated with a group word can be solved over a hyperlinear group if its content — that is, its augmentation in — does not lie in the second term of the lower central series of . Moreover, if is finite, then a solution can be found in a finite extension of . The method of proof extends techniques developed by Gerstenhaber and Rothaus, and uses computations in –local homotopy theory and cohomology of compact Lie groups.
A –local compact group is an algebraic object modelled on the homotopy theory associated with –completed classifying spaces of compact Lie groups and –compact groups. In particular –local compact groups give a unified framework in which one may study –completed classifying spaces from an algebraic and homotopy theoretic point of view. Like connected compact Lie groups and –compact groups, –local compact groups admit unstable Adams operations: self equivalences that are characterised by their cohomological effect. Unstable Adams operations on –local compact groups were constructed in a previous paper by F Junod, R Levi, and A Libman. In the present paper we study groups of unstable operations from a geometric and algebraic point of view. We give a precise description of the relationship between algebraic and geometric operations, and show that under some conditions, unstable Adams operations are determined by their degree. We also examine a particularly well behaved subgroup of unstable Adams operations.
If is a finite lattice, we show that there is a natural topological lattice structure on the geometric realization of its order complex (definition recalled below). Lattice-theoretically, the resulting object is a subdirect product of copies of . We note properties of this construction and of some variants, and pose several questions. For the –element nondistributive modular lattice, is modular, but its underlying topological space does not admit a structure of distributive lattice, answering a question of Walter Taylor.
We also describe a construction of “stitching together” a family of lattices along a common chain, and note how can be regarded as an example of this construction.
We consider the question of which Dehn surgeries along a given knot bound rational homology balls. We use Ozsváth and Szabó’s correction terms in Heegaard Floer homology to obtain general constraints on the surgery coefficients. We then turn our attention to the case of integral surgeries, with particular emphasis on positive torus knots. Finally, combining these results with a lattice-theoretic obstruction based on Donaldson’s theorem, we classify which integral surgeries along torus knots of the form bound rational homology balls.
We show that every countable subgroup without contracting elements is the Veech group of a tame translation surface of infinite genus for infinitely many different topological types of . Moreover, we prove that as long as every end has genus, there are no restrictions on the topological type of to realize all possible uncountable Veech groups.
The purpose of this note is to answer Question 6.12 of Etnyre and Van Horn-Morris [Monoids in the mapping class group, Geom. Topol. Monographs 19 (2015) 319–365], asking when the set of mapping classes whose fractional Dehn twist coefficient is greater than a given constant forms a monoid.
Our goal in this paper is to prove an equivalence between the model categorical approach to pro-categories, as studied by Isaksen, Schlank and the first author, and the –categorical approach, as developed by Lurie. Three applications of our main result are described. In the first application we use (a dual version of) our main result to give sufficient conditions on an –combinatorial model category, which insure that its underlying –category is –presentable. In the second application we show that the topological realization of any Grothendieck topos coincides with the shape of the hypercompletion of the associated –topos. In the third application we show that several model categories arising in profinite homotopy theory are indeed models for the –category of profinite spaces. As a byproduct we obtain new Quillen equivalences between these models, and also obtain an example which settles negatively a question raised by G Raptis.
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