Journal of Topology最新文献

We consider the set of connected surfaces in the 4-ball with boundary a fixed knot in the 3-sphere. We define the stabilization distance between two surfaces as the minimal $�g$ such that we can get from one to the other using stabilizations and destabilizations through surfaces of genus at most $�g$. Similarly, we consider a double-point distance between two surfaces of the same genus that is the minimum over all regular homotopies connecting the two surfaces of the maximal number of double points appearing in the homotopy. To many of the concordance invariants defined using Heegaard Floer homology, we construct an analogous invariant for a pair of surfaces. We show that these give lower bounds on the stabilization distance and the double-point distance. We compute our invariants for some pairs of deform-spun slice disks by proving a trace formula on the full infinity knot Floer complex, and by determining the action on knot Floer homology of an automorphism of the connected sum of a knot with itself that swaps the two summands. We use our invariants to find pairs of slice disks with arbitrarily large distance with respect to many of the metrics we consider in this paper. We also answer a slice-disk analog of Problem 1.105 (B) from Kirby's problem list by showing the existence of non-0-cobordant slice disks.

Knot filtered embedded contact homology was first introduced by Hutchings in 2015; it has been computed for the standard transverse unknot in irrational ellipsoids by Hutchings and for the Hopf link in lens spaces $��L�(�n�,�n�-�1�)�$ via a quotient by Weiler. While toric constructions can be used to understand the ECH chain complexes of many contact forms adapted to open books with binding the unknot and Hopf link, they do not readily adapt to general torus knots and links. In this paper, we generalize the definition and invariance of knot filtered embedded contact homology to allow for degenerate knots with rational rotation numbers. We then develop new methods for understanding the embedded contact homology chain complex of positive torus knotted fibrations of the standard tight contact 3-sphere in terms of their presentation as open books and as Seifert fiber spaces. We provide Morse–Bott methods, using a doubly filtered complex and the energy filtered perturbed Seiberg–Witten Floer theory developed by Hutchings and Taubes, and use them to compute the $��T�(�2�,�q�)�$ knot filtered embedded contact homology, for $�q$ odd and positive.

For any smooth manifold $�M$ of dimension $��d�⩾�4�$, we construct explicit classes in homotopy groups of spaces of embeddings of either an arc or a circle into $�M$, in every degree that is a multiple of $��d�-�3�$, and show that they are detected in the Taylor tower of Goodwillie and Weiss. The classes are obtained from families of string links constructed in the $�d$-ball.

We develop tools to analyze and compare the Brauer groups of spectra such as periodic complex and real $�K$-theory and topological modular forms, as well as the derived moduli stack of elliptic curves. In particular, we prove that the Brauer group of $�\mathrm{TMF}$ is isomorphic to the Brauer group of the derived moduli stack of elliptic curves. Our main computational focus is on the subgroup of the Brauer group consisting of elements trivialized by some étale extension, which we call the local Brauer group. Essential information about this group can be accessed by a thorough understanding of the Picard sheaf and its cohomology. We deduce enough information about the Picard sheaf of $�\mathrm{TMF}$ and the (derived) moduli stack of elliptic curves to determine the structure of their local Brauer groups away from the prime 2. At 2, we show that they are both infinitely generated and agree up to a potential error term that is a finite 2-torsion group.

A representation of a finitely generated group into the projective general linear group is called convex co-compact if it has finite kernel and its image acts convex co-compactly on a properly convex domain in real projective space. We prove that the fundamental group of a closed irreducible orientable 3-manifold can admit such a representation only when the manifold is geometric (with Euclidean, Hyperbolic or Euclidean $�\times$ Hyperbolic geometry) or when every component in the geometric decomposition is hyperbolic. In each case, we describe the structure of such examples.

We show that Koszul duality for operads in $��(�\mathrm{Top}�,�\times �)�$ can be expressed via generalized Thom complexes. As an application, we prove the Koszul self-duality of the right module $�E_M$ associated to a framed manifold $�M$. We discuss implications for factorization homology, embedding calculus, and confirm an old conjecture of Ching on the relation of Goodwillie calculus to manifold calculus.

Let $�L$ be an exact Lagrangian submanifold of a cotangent bundle $��T^{\ast }�M�$, asymptotic to a Legendrian submanifold $��\Lambda �\subset �T^{\infty }�M�$. We study a locally constant sheaf of $�\infty$-categories on $�L$, called the sheaf of brane structures or $�{\mathrm{Brane}}_L$. Its fiber is the $�\infty$-category of spectra, and we construct a Hamiltonian invariant, fully faithful functor from $��\Gamma �(�L�,�{\mathrm{Brane}}_L�)�$ to the $�\infty$-category of sheaves of spectra on $�M$ with singular support in $�\Lambda$.

Given a class of embeddings into a contact or a symplectic manifold, we give a sufficient condition, that we call isocontact or isosymplectic realization, for this class to satisfy a general $�h$-principle. The flexibility follows from the $�h$-principles for isocontact and isosymplectic embeddings, it provides a framework for classical results, and we give two new applications. Our main result is that embeddings transverse to a contact structure satisfy a full $�h$-principle in two cases: if the complement of the embedding is overtwisted, or when the intersection of the image of the formal derivative with the contact structure is strictly contained in a proper symplectic subbundle. We illustrate the general framework on symplectic manifolds by studying the universality of Hamiltonian dynamics on regular level sets via a class of embeddings.

We study several aspects of fillings for links of general isolated quotient singularities using Floer theory, including co-fillings, Weinstein fillings, strong fillings, exact fillings and exact orbifold fillings, focusing on the non-existence of exact fillings of contact links of isolated terminal quotient singularities. We provide an extensive list of isolated terminal quotient singularities whose contact links are not exactly fillable, including $��C^n�/��(�Z�/�2�)��$ for $��n�⩾�3�$, which settles a conjecture of Eliashberg, quotient singularities from general cyclic group actions and finite subgroups of $��S�U�\left(�2�\right)�$, and all terminal quotient singularities in complex dimension 3. We also obtain uniqueness of the orbifold diffeomorphism type of exact orbifold fillings of contact links of some isolated terminal quotient singularities.