Journal of Topology最新文献

We give a computability result for open Gromov–Witten invariants based on open Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations. This is analogous to the result of Kontsevich–Manin for closed Gromov–Witten invariants. For greater generality, we base the argument on a formal object, the Frobenius superpotential, that generalizes several different definitions of open Gromov–Witten invariants. As an application, we prove vanishing properties for open Gromov–Witten invariants on products of projective spaces.

The original Arnold chord conjecture states that every closed Legendrian submanifold of the standard contact sphere $�S^{�2�n�-�1�}$ admits a Reeb chord with distinct endpoints with respect to any contact form. In this paper, we prove this conjecture for contact forms induced by strictly convex embeddings into $�R^{�2�n�}$ under the assumption that minimal periodic Reeb orbits are of Morse–Bott type. We also provide a counterexample when the convexity condition is not satisfied.

Let $�\phi$ be a pseudo-Anosov flow on a closed oriented atoroidal 3-manifold $�M$. We show that if $�F$ is any taut foliation almost transverse to $�\phi$, then the action of $��{\pi }_1��\left(�M�\right)��$ on the boundary of the flow space, together with a natural collection of explicitly described monotone maps, defines a universal circle for $�F$ in the sense of Thurston and Calegari–Dunfield.

We prove an $�\infty$-categorical version of the reflection theorem of AdÁmek and Rosický [Arch. Math. 25 (1989), no. 1, 89–94]. Namely, that a full subcategory of a presentable $�\infty$-category that is closed under limits and $�\kappa$-filtered colimits is a presentable $�\infty$-category. We then use this theorem in order to classify subcategories of a symmetric monoidal $�\infty$-category that are equivalent to a category of modules over an idempotent algebra.

We construct a Lagrangiansubmanifold in the cotangent bundle of a 3-torus whose projection to the fiber is a neighborhood of a tropical curve with a single 4-valent vertex. This Lagrangian submanifold has an isolated conical singular point, and its smooth locus is diffeomorphic to the minimally twisted five-component chain link complement, a cusped hyperbolic 3-manifold. From this singular Lagrangian, we construct a Lagrangian immersion, and identify a moduli space of objects in the wrapped Fukaya category that it supports. We show that for a generic line in projective 3-space, there is a local system on this Lagrangian immersion such that the resulting object of the wrapped Fukaya category is homologically mirror to an object of the derived category supported on the line. In the course of the proof, we construct a version of the wrapped Fukaya category with objects supported on Lagrangian immersions, which may be of independent interest.

A diffeomorphism $�f$ of a compact manifold $�X$ is pseudo-isotopic to the identity if there is a diffeomorphism $�F$ of $��X�\times �I�$ which restricts to $�f$ on $��X�\times �1�$, and which restricts to the identity on $��X�\times �0�$ and $��\partial �X�\times �I�$. We construct examples of diffeomorphisms of 4-manifolds which are pseudo-isotopic but not isotopic to the identity. To do so, we further understanding of which elements of the ‘second pseudo-isotopy obstruction’, defined by Hatcher and Wagoner, can be realised by pseudo-isotopies of 4-manifolds. We also prove that all elements of the first and second pseudo-isotopy obstructions can be realised after connected sums with copies of $��S^2�\times �S^2�$.

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.