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We functorially associate to each relative –category a simplicial space , called its Rezk nerve (a straightforward generalization of Rezk’s “classification diagram” construction for relative categories). We prove the following local and global universal properties of this construction: (i) that the complete Segal space generated by the Rezk nerve is precisely the one corresponding to the localization ; and (ii) that the Rezk nerve functor defines an equivalence from a localization of the –category of relative –categories to the –category of –categories.
This paper studies how symplectic invariants created from Hamiltonian Floer theory change under the perturbations of symplectic structures, notnecessarily in the same cohomology class. These symplectic invariants include spectral invariants, boundary depth, and (partial) symplectic quasistates. This paper can split into two parts. In the first part, we prove some energy estimations which control the shifts of symplectic action functionals. These directly imply positive conclusions on the continuity of spectral invariants and boundary depth in some important cases, including any symplectic surface and any closed symplectic manifold with . This follows by applications on some rigidity of the subsets of a symplectic manifold in terms of heaviness and superheaviness, as well as on the continuity property of some symplectic capacities. In the second part, we generalize the construction in the first part to any closed symplectic manifold. In particular, to deal with the change of Novikov rings from symplectic structure perturbations, we construct a family of variant Floer chain complexes over a common Novikov-type ring. In this setup, we define a new family of spectral invariants called –spectral invariants, and prove that they are upper semicontinuous under the symplectic structure perturbations. This implies a quasi-isometric embedding from to under some dynamical assumption, imitating the main result of Usher (Ann. Sci. Éc. Norm. Supér. 46 (2013) 57–128).
To a region of the plane satisfying a suitable convexity condition we associate a knot concordance invariant . For appropriate choices of the domain this construction gives back some known knot Floer concordance invariants like Rasmussen’s invariants, and the Ozsváth–Stipsicz–Szabó upsilon invariant. Furthermore, to three such regions , and we associate invariants generalizing the Kim–Livingston secondary invariant. We show how to compute these invariants for some interesting classes of knots (including alternating and torus knots), and we use them to obstruct concordances to Floer thin knots and algebraic knots.
Ropelength and embedding thickness are related measures of geometric complexity of classical knots and links in Euclidean space. In their recent work, Freedman and Krushkal posed a question regarding lower bounds for embedding thickness of –component links in terms of the Milnor linking numbers. The main goal of the current paper is to provide such estimates, and thus generalize the known linking number bound. In the process, we collect several facts about finite-type invariants and ropelength/crossing number of knots. We give examples of families of knots where such estimates behave better than the well-known knot–genus estimate.
We show rational homological stability for the classifying spaces of the monoid of homotopy self-equivalences and the block diffeomorphism group of iterated connected sums of products of spheres. The spheres can have different dimensions, but need to satisfy a certain connectivity assumption. The main theorems of this paper extend homological stability results for automorphism spaces of connected sums of products of spheres of the same dimension by Berglund and Madsen.
The Bourgeois construction associates to every contact open book on a manifold a contact structure on . We study some of the properties of that are inherited by and some that are not.
Giroux has provided recently a better framework to work with contact open books. In the appendix, we quickly review this formalism, and we work out a few classical examples of contact open books to illustrate how to use this new language.
We study the skein algebras of marked surfaces and the skein modules of marked –manifolds. Muller showed that skein algebras of totally marked surfaces may be embedded in easy-to-study algebras known as quantum tori. We first extend Muller’s result to permit marked surfaces with unmarked boundary components. The addition of unmarked components allows us to develop a surgery theory which enables us to extend the Chebyshev homomorphism of Bonahon and Wong between skein algebras of unmarked surfaces to a “Chebyshev–Frobenius homomorphism” between skein modules of marked –manifolds. We show that the image of the Chebyshev–Frobenius homomorphism is either transparent or skew-transparent. In addition, we make use of the Muller algebra method to calculate the center of the skein algebra of a marked surface when the quantum parameter is not a root of unity.
For links with vanishing pairwise linking numbers, the link components bound pairwise disjoint oriented surfaces in . We use the –function which is a link invariant from the Heegaard Floer homology to give lower bounds for the –genus of the link. For –space links, the –function is explicitly determined by the Alexander polynomials of the link and its sublinks. We show some –space links where the lower bounds are sharp, and also describe all possible genera of disjoint oriented surfaces bounded by such links.
For a compact Lie group, toral –spectra are those rational –spectra whose geometric isotropy consists of subgroups of a maximal torus of . The homotopy category of rational toral –spectra is a retract of the category of all rational –spectra.
We show that the abelian category of Greenlees (Algebr. Geom. Topol. 16 (2016) 1953–2019) gives an algebraic model for the toral part of rational –spectra. This is a major step in establishing an algebraic model for all rational –spectra for any compact Lie group .
We study the local homology of Artin groups using weighted discrete Morse theory. In all finite and affine cases, we are able to construct Morse matchings of a special type (we call them “precise matchings”). The existence of precise matchings implies that the homology has a squarefree torsion. This property was known for Artin groups of finite type, but not in general for Artin groups of affine type. We also use the constructed matchings to compute the local homology in all exceptional cases, correcting some results in the literature.
Let be a field with a real embedding. We compare the motivic slice filtration of a motivic spectrum over with the –equivariant slice filtration of its equivariant Betti realization, giving conditions under which realization induces an equivalence between the associated slice towers. In particular, we show that, up to reindexing, the towers agree for all spectra obtained from localized quotients of and , and for motivic Landweber exact spectra and their realizations. As a consequence, we deduce that equivariant spectra obtained from localized quotients of are even in the sense of Hill and Meier, and give a computation of the slice spectral sequence converging to for .
We show that the class and the LOSS invariant of Legendrian knots in contact –manifolds are functorial under regular Lagrangian concordances in Weinstein cobordisms. This gives computable obstructions to the existence of regular Lagrangian concordances.
Let be the fundamental group of a finite graph of groups with Noetherian edge groups and locally tame vertex groups. We prove that is locally tame. It follows that if a finitely presented group has a nontrivial –decomposition over the class of its subgroups for or , and all the vertex groups in the decomposition are flexible, then is locally tame.
Tensoring finite pointed simplicial sets with commutative ring spectra yields important homology theories such as (higher) topological Hochschild homology and torus homology. We prove several structural properties of these constructions relating to and we establish splitting results. This allows us, among other important examples, to determine for all and for all .
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