Journal of Topology最新文献

We unify two existing approaches to the tau invariants in instanton and monopole Floer theories, by identifying $�{\tau }_G$, defined by the second author via the minus flavors $�{\underline{KHI}}^{-}$ and $�{\underline{KHM}}^{-}$ of the knot homologies, with $�{\tau }_G^{♯}$, defined by Baldwin and Sivek via cobordism maps of the 3-manifold homologies induced by knot surgeries. We exhibit several consequences, including a relationship with Heegaard Floer theory, and use our result to compute $�{\underline{KHI}}^{-}$ and $�{\underline{KHM}}^{-}$ for twist knots.

In this paper we further develop the theory of equivariant Seiberg–Witten–Floer cohomology of the two authors, with an emphasis on Brieskorn homology spheres. We obtain a number of applications. First, we show that the knot concordance invariants $�{\theta }^{�\left(�c�\right)�}$ defined by the first author satisfy $��{\theta }^{�\left(�c�\right)�}��\left(�T_{�a�,�b�}�\right)��=��(�a�-�1�)���(�b�-�1�)��/�2�$ for torus knots, whenever $�c$ is a prime not dividing $��a�b�$. Since $�{\theta }^{�\left(�c�\right)�}$ is a lower bound for

Proposition 6.9 in (J. Topol. 15 (2022), no. 4, 2270–2297) is incorrect without a connectivity assumption. In this note, we provide a counter-example, give a correct proof of the modified proposition and explain the other changes that need to be made to [1].

We give an expression for the Smith–Thom deficiency of the Hilbert square $�X^{�\left[�2�\right]�}$ of a smooth real algebraic variety $�X$ in terms of the rank of a suitable Mayer– Vietoris mapping in several situations. As a consequence, we establish a necessary and sufficient condition for the maximality of $�X^{�\left[�2�\right]�}$ in the case of projective complete intersections, and show that with a few exceptions, no real nonsingular projective complete intersection of even dimension has maximal Hilbert square. We also provide new examples of smooth real algebraic varieties with maximal Hilbert square.

A $�d$-dimensional invertible topological field theory (TFT) is a functor from the symmetric monoidal $��(�\infty �,�n�)�$-category of $�d$-bordisms (embedded into $�R^{\infty }$ and equipped with a tangential $��(�X�,�\xi �)�$-structure) that lands in the Picard subcategory of the target symmetric monoidal $��(�\infty �,�n�)�$-category. We classify these field theories in terms of the cohomology of the $��(�n�-�d�)�$-connective cover of the Madsen–Tillmann spectrum. This is accomplished by identifying the classifying space of the $��(�\infty �,�n�)�$-category of bordisms with $��{\Omega }^{�\infty �-�n�}�M�T�\xi �$

In a 2005 paper, Casacuberta, Scevenels, and Smith construct a homotopy idempotent functor $�E$ on the category of simplicial sets with the property that whether it can be expressed as localization with respect to a map $�f$ is independent of the ZFC axioms. We show that this construction can be carried out in homotopy type theory. More precisely, we give a general method of associating to a suitable (possibly large) family of maps, a reflective subuniverse of any universe $�U$. When specialized to an appropriate family, this produces a localization which when interpreted in the $�\infty$-topos of spaces agrees with the localization corresponding to $�E$. Our approach generalizes the approach of Casacuberta et al. (Adv. Math. 197 (2005), no. 1, 120–139) in two ways. First, by working in homotopy type theory, our construction can be interpreted in any $�\infty$-topos. Second, while the local objects produced by Casacuberta et al. are always 1-types, our construction can produce $�n$-types, for any $�n$. This is new, even in the $�\infty$-topos of spaces. In addition, by making use of universes, our proof is very direct. Along the way, we prove many results about “small” types that are of independent interest. As an application, we give a new proof that separated localizations exist. We also give results that say when a localization with respect to a family of maps can be presented as localization with respect to a single map, and show that the simplicial model satisfies a strong form of the axiom of choice that implies that sets cover and that the law of excluded middle holds.

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.