Journal of Topology最新文献

We exhibit infinitely many ribbon knots, each of which bounds infinitely many pairwise nonisotopic ribbon disks whose exteriors are diffeomorphic. This family provides a positive answer to a stronger version of an old question of Hitt and Sumners. The examples arise from our main result: a classification of fibered, homotopy-ribbon disks for each generalized square knot $��T_{�p�,�q�}�\#�{\overline{T}}_{�p�,�q�}�$ up to isotopy. Precisely, we show that each generalized square knot bounds infinitely many pairwise nonisotopic fibered, homotopy-ribbon disks, all of whose exteriors are diffeomorphic. When $��q�=�2�$, we prove further that infinitely many of these disks are also ribbon; whether the disks are always ribbon is an open problem.

The variation operator in singularity theory maps relative homology cycles to compact cycles in the Milnor fiber using the monodromy. We construct its symplectic analog for an isolated singularity. We define the monodromy Lagrangian Floer cohomology, which provides categorifications of the standard theorems on the variation operator and the Seifert form. The key ingredients are a special class $�\Gamma$ in the symplectic cohomology of the inverse of the monodromy and its closed–open images. For isolated plane curve singularities whose A'Campo divide has depth zero, we find an exceptional collection consisting of noncompact Lagrangians in the Milnor fiber corresponding to a distinguished collection of vanishing cycles under the variation operator.

In a previous paper, we introduced quasi-BPS categories for moduli stacks of semistable Higgs bundles. Under a certain condition on the rank, Euler characteristic, and weight, the quasi-BPS categories (called BPS in this case) are noncommutative analogues of Hitchin integrable systems. We proposed a conjectural equivalence between BPS categories which swaps Euler characteristics and weights. The conjecture is inspired by the Dolbeault Geometric Langlands equivalence of Donagi–Pantev, by the Hausel–Thaddeus mirror symmetry, and by the $�\chi$-independence phenomenon for BPS invariants of curves on Calabi–Yau threefolds. In this paper, we show that the above conjecture holds at the level of topological K-theories. When the rank and the Euler characteristic are coprime, such an isomorphism was proved by Groechenig–Shen. Along the way, we show that the topological K-theory of BPS categories is isomorphic to the BPS cohomology of the moduli of semistable Higgs bundles.

We develop explicit local operations that may be applied to Liouville domains, with the goal of simplifying the dynamics of the Liouville vector field. These local operations, which are Liouville homotopies, are inspired by the techniques used by Honda and Huang [arXiv:1907.06025, 2019] to show that convex hypersurfaces are $�C^0$-generic in contact manifolds. As an application, we use our operations to show that certain Liouville-but-not-Weinstein domains constructed by Huang [Proc. Amer. Math. Soc. 148 (2020), no. 12, 5323–5330] are stably Weinstein.

In this paper, we show that the six functor formalism for sheaves on locally compact Hausdorff topological spaces, as developed, for example,- in Kashiwara and Schapira's book Sheaves on Manifolds, can be extended to sheaves with values in any closed symmetric monoidal $�\infty$-category which is stable and bicomplete. Notice that, since we do not assume that our coefficients are presentable or restrict to hypercomplete sheaves, our arguments are not obvious and are substantially different from the ones explained by Kashiwara and Schapira. Along the way we also study locally contractible geometric morphisms and prove that, if $��f�:�X�\to �Y�$ is a continuous map which induces a locally contractible geometric morphism, then the exceptional pullback functor $�f^{!}$ preserves colimits and can be related to the pullback $�f^{*}$. At the end of our paper, we also show how one can express Atiyah duality by means of the six functor formalism.

We develop versions of the Poincaré–Birkhoff–Witt and Cartier–Milnor–Moore theorems in the setting of braided Hopf algebras. To do so, we introduce new analogs of a Lie algebra in the setting of a braided monoidal category, using the notion of a braided operad.

We prove the existence of a subclass of overtwisted contact structures, called strongly overtwisted, on a 3-manifold that satisfy a complete $�h$-principle without prescribing the contact structures over any subset of the 3-manifold. As a consequence, the homotopy type of the space of overtwisted disk embeddings into a strongly overtwisted contact 3-manifold is determined. A complete $�h$-principle for a subclass of loose Legendrians is also derived from the main result. In general, the method allows us to deduce an $�h$-principle for overtwisted disks that are fixed near the boundary in an arbitrary overtwisted contact 3-manifold.

We develop a framework of parametrized semiadditivity and stability with respect to so-called atomic orbital subcategories of an indexing $�\infty$-category $�T$, extending work of Nardin. Specializing this framework, we introduce global $�\infty$-categories and the notions of equivariant semiadditivity and stability, yielding a higher categorical version of the notion of a Mackey 2-functor studied by Balmer–Dell'Ambrogio. As our main result, we identify the free presentable equivariantly stable global $�\infty$-category with a natural global $�\infty$-category of global spectra for finite groups, in the sense of Schwede and Hausmann.

Every geodesic current on a hyperbolic surface has an associated dual space. If the current is a lamination, this dual embeds isometrically into a real tree. We show that, in general, the dual space is a Gromov hyperbolic metric tree-graded space, and express its Gromov hyperbolicity constant in terms of the geodesic current. In the case of geodesic currents with no atoms and full support, such as those coming from certain higher rank representations, we show the duals are homeomorphic to the surface. We also analyze the completeness of the dual and the properties of the action of the fundamental group of the surface on the dual. Furthermore, we compare two natural topologies in the space of duals.

We study the action of the orthogonal group on the little $�n$-disks operads. As an application we provide small models (over the reals) for the framed little $�n$-disks operads. It follows in particular that the framed little $�n$-disks operads are formal (over the reals) for $�n$ even and coformal for all $�n$.