Journal of Topology最新文献

We consider dynamically convex star-shaped domains in a symplectic vector space of dimension 4. For such a domain, a “Hopf orbit” is a closed characteristic in the boundary which is unknotted and has self-linking number $��-�1�$. We show that the minimum action among Hopf orbits exists and defines a symplectic capacity for dynamically convex star-shaped domains. We further show that this capacity agrees with the first embedded contact homology (ECH) capacity for such domains. Combined with a result of Edtmair, this implies that for dynamically convex star-shaped domains in four dimensions, the first ECH capacity agrees with the cylinder capacity. This also provides a method to show that the first ECH capacity of a dynamically convex star-shaped domain satisfies the axioms of a normalized symplectic capacity without any need for Seiberg–Witten theory.

We introduce the notion of graphical discreteness to group theory. A finitely generated group is graphically discrete if whenever it acts geometrically on a locally finite graph, the automorphism group of the graph is compact-by-discrete. Notable examples include finitely generated nilpotent groups, most lattices in semisimple Lie groups, and irreducible nongeometric 3-manifold groups. We show graphs of groups with graphically discrete vertex groups frequently have strong rigidity properties. We prove free products of one-ended virtually torsion-free graphically discrete groups are action rigid within the class of virtually torsion-free groups. We also prove quasi-isometric rigidity for many hyperbolic graphs of groups whose vertex groups are closed hyperbolic manifold groups and whose edge groups are nonelementary quasi-convex subgroups. This includes the case of two hyperbolic 3-manifold groups amalgamated along a quasi-convex malnormal non-abelian free subgroup. We provide several additional examples of graphically discrete groups and illustrate this property is not a commensurability invariant.

Inspired by the notions of local equivalence in monopole and Heegaard Floer homology, we introduce a version of local equivalence that combines odd Khovanov homology with equivariant even Khovanov homology into an algebraic package called a local even–odd (LEO) triple. We get a homomorphism from the smooth concordance group $�C$ to the resulting local equivalence group $�C_{\mathrm{LEO}}$ of such triples. We give several versions of the $�s$-invariant that descend to $�C_{\mathrm{LEO}}$, including one that completely determines whether the image of a knot $�K$ in $�C_{\mathrm{LEO}}$ is trivial. We discuss computer experiments illustrating the power of these invariants in obstructing sliceness, both statistically and for some interesting knots studied by Manolescu–Piccirillo. Along the way, we explore several variants of this local equivalence group, including one that is totally ordered.

We establish an entropy rigidity theorem for Hitchin representations of geometrically finite Fuchsian groups which generalizes a theorem of Potrie and Sambarino for Hitchin representations of closed surface groups. In the process, we introduce the class of (1,1,2)-hypertransverse groups and show for such a group that the Hausdorff dimension of its conical limit set agrees with its (first) simple root entropy, providing a common generalization of results of Bishop and Jones, for Kleinian groups, and Pozzetti, Sambarino, and Wienhard, for Anosov groups. We also introduce the theory of transverse representations of projectively visible groups as a tool for studying discrete subgroups of linear groups that are not necessarily Anosov or relatively Anosov.

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.