Journal of Topology and Analysis

We (re)consider how the Fukaya category of a Lefschetz fibration is related to that of the fiber. The distinguishing feature of the approach here is a more direct identification of the bimodule homomorphism involved.

A Finsler manifold is said to be geodesically reversible if the reversed curve of any geodesic remains a geometrical geodesic. Well-known examples of geodesically reversible Finsler metrics are Randers metrics with closed $1$-forms. Another family of well-known examples are projectively flat Finsler metrics on the $2$-sphere that have constant positive curvature. In this paper, we prove some geometrical and dynamical characterizations of geodesically reversible Finsler metrics, and we prove several rigidity results about a family of the so-called Randers-type Finsler metrics. One of our results is as follows: let $\sqrt{g}$ be a Riemannian–Finsler metric on a closed surface $\mathrm{\Sigma }$, and $p$ be a small antisymmetric potential on $\mathrm{\Sigma }$ that is a natural generalization of $1$-form (see Sec. 1). If the Randers-type Finsler metric $\sqrt{g}-p$ is geodesically reversible, and the geodesic flow of $\sqrt{g}$ is topologically transitive, then we prove that $p$ must be a closed $1$-form. We also prove that this rigidity result is not true for the family of projectively flat Finsler metrics on the $2$-sphere of constant positive curvature.

This paper is a systematic approach to the construction of coronas (i.e. Higson dominated boundaries at infinity) of combable spaces. We introduce three additional properties for combings: properness, coherence and expandingness. Properness is the condition under which our construction of the corona works. Under the assumption of coherence and expandingness, attaching our corona to a Rips complex construction yields a contractible $\sigma$-compact space in which the corona sits as a $Z$-set. This results in bijectivity of transgression maps, injectivity of the coarse assembly map and surjectivity of the coarse co-assembly map. For groups we get an estimate on the cohomological dimension of the corona in terms of the asymptotic dimension. Furthermore, if the group admits a finite model for its classifying space $BG$, then our constructions yield a $Z$-structure for the group.

We prove that a Morse–Smale gradient-like flow on a closed manifold has a “system of compatible invariant stable foliations” that is analogous to the object introduced by Palis and Smale in their proof of the structural stability of Morse–Smale diffeomorphisms and flows, but with finer regularity and geometric properties. We show how these invariant foliations can be used in order to give a self-contained proof of the well-known but quite delicate theorem stating that the unstable manifolds of a Morse–Smale gradient-like flow on a closed manifold $M$ are the open cells of a CW-decomposition of $M$.

An oriented equator of ${𝕊}^2$ is the image of an oriented embedding ${𝕊}^1\hookrightarrow {𝕊}^2$ such that it divides ${𝕊}^2$ into two equal area halves. Following Chekanov, we define the Hofer distance between two oriented equators as the infimal Hofer norm of a Hamiltonian diffeomorphism taking one to another. Consider ${𝜖}_{q+}$ the space of oriented equators. We define the Hofer girth of an embedding $j:{𝕊}^2\hookrightarrow {𝜖}_{q+}$ as the infimum of the Hofer diameter of $j^{\prime }({𝕊}^2)$, where $j^{\prime }$ is homotopic to $j$. There is a natural embedding $i_0:{𝕊}^2\hookrightarrow {𝜖}_{q+}$, sending a point on the sphere to the positively oriented great circle perpendicular to it. In this paper, we provide an upper bound on the Hofer girth of $i_0$.

By computing the completely bounded norm of the flip map on the Haagerup tensor product $C_0{Y}_1{\otimes }_{C_{0}X}{C}_0{Y}_2$ associated to a pair of continuous mappings of locally compact Hausdorff spaces $Y_1\to X\leftarrow Y_2$, we establish a simple characterization of the Beck-Chevalley condition for base change of operator modules over commutative $C^{\ast }$-algebras, and a descent theorem for continuous fields of Hilbert spaces.

The notion of parametrized topological complexity, introduced by Cohen, Farber and Weinberger, is extended to fiberwise spaces which are not necessarily Hurewicz fibrations. After exploring some formal properties of this extension we also introduce the pointed version of parametrized topological complexity. Finally, we give sufficient conditions so that both notions agree.