Journal of Topology and Analysis

We relate the quantum Steenrod square to Seidel’s equivariant pair-of-pants product for open convex symplectic manifolds that are either monotone or exact, using an equivariant version of the PSS isomorphism. We define continuation maps between different Hamiltonians in $\mathbb{Z}/2$-equivariant Floer cohomology, and prove expected properties of them. We prove a symplectic Cartan relation for the equivariant pair-of-pants product, pointing out the difficulties in stating it. We give a nonvanishing result for the equivariant pair-of-pants product for some elements of $S{H}^{\ast }(T^{\ast }{S}^n)$. We finish by calculating the symplectic square for the negative line bundles $M=\text{Tot}({𝒪}(-1)\to {ℂ\mathbb{P}}^m)$, proving an equivariant version of a result due to Ritter.

This paper uses relative symplectic cohomology, recently studied by Varolgunes, to understand rigidity phenomena for compact subsets of symplectic manifolds. As an application, we consider a symplectic crossings divisor in a Calabi–Yau symplectic manifold $M$ whose complement is a Liouville manifold. We show that, for a carefully chosen Liouville structure, the skeleton as a subset of $M$ exhibits strong rigidity properties akin to superheavy subsets of Entov–Polterovich. Along the way, we expand the toolkit of relative symplectic cohomology by introducing products and units. We also develop what we call the contact Fukaya trick, concerning the behavior of relative symplectic cohomology of subsets with contact type boundary under adding a Liouville collar.

Suppose a finitely generated group $G$ is hyperbolic relative to ${𝒫}$ a set of proper finitely generated subgroups of $G$. Established results in the literature imply that a “visual” metric on $\partial (G,{𝒫})$ is “linearly connected” if and only if the boundary $\partial (G,{𝒫})$ has no cut point. Our goal is to produce linearly connected metrics on $\partial (G,{𝒫})$ that are “piecewise” visual when $\partial (G,{𝒫})$ contains cut points. Our main theorem is connected to graph of groups decompositions of relatively hyperbolic groups $(G,{𝒫})$ by work of B. Bowditch. We describe piecewise visual linearly connected metrics on connected boundaries of relatively hyperbolic groups. Our metric on $\partial (G,{𝒫})$ agrees with the visual metric on limit sets of vertex groups and is in this sense piecewise visual.

We show that the integral foliated simplicial volume of a compact oriented smooth manifold with a regular foliation by circles vanishes.

In this paper, we introduce a fast and memory efficient approach to compute the Persistent Homology (PH) of a sequence of simplicial complexes. The basic idea is to simplify the complexes of the input sequence by using strong collapses, as introduced by Barmak and Miniam [DCG (2012)], and to compute the PH of an induced sequence of reduced simplicial complexes that has the same PH as the initial one. Our approach has several salient features that distinguishes it from previous work. It is not limited to filtrations (i.e. sequences of nested simplicial subcomplexes) but works for other types of sequences like towers and zigzags. To strong collapse a simplicial complex, we only need to store the maximal simplices of the complex, not the full set of all its simplices, which saves a lot of space and time. Moreover, the complexes in the sequence can be strong collapsed independently and in parallel.

We also focus on the problem of computing persistent homology of a flag tower, i.e. a sequence of flag complexes connected by simplicial maps. We show that if we restrict the class of simplicial complexes to flag complexes, we can achieve decisive improvement in terms of time and space complexities with respect to previous work. Moreover we can strong collapse a flag complex knowing only its 1-skeleton and the resulting complex is also a flag complex. When we strong collapse the complexes in a flag tower, we obtain a reduced sequence that is also a flag tower we call the core flag tower. We then convert the core flag tower to an equivalent filtration to compute its PH. Here again, we only use the 1-skeletons of the complexes. The resulting method is simple and extremely efficient. As a result and as demonstrated by numerous experiments on publicly available data sets, our approach is extremely fast and memory efficient in practice. Finally, we can compromise between precision and time by choosing the number of simplicial complexes of the sequence we strong collapse.

We show that the Torelli group of a closed surface of genus $\ge 3$ acts nontrivially on the rational cohomology of the space of $3$-element subsets of that surface. This implies that for a Riemann surface of genus $\ge 3$, the mixed Hodge structure on the space of its positive, reduced divisors of degree $3$ does in general not split over $\mathbb{Q}$.

This paper is part of the program of studying large-scale geometric properties of totally disconnected locally compact groups, TDLC-groups, by analogy with the theory for discrete groups. We provide a characterization of hyperbolic TDLC-groups, in terms of homological isoperimetric inequalities. This characterization is used to prove the main result of this paper: for hyperbolic TDLC-groups with rational discrete cohomological dimension $\le 2$, hyperbolicity is inherited by compactly presented closed subgroups. As a consequence, every compactly presented closed subgroup of the automorphism group $Aut(X)$ of a negatively curved locally finite $2$-dimensional building $X$ is a hyperbolic TDLC-group, whenever $Aut(X)$ acts with finitely many orbits on $X$. Examples where this result applies include hyperbolic Bourdon’s buildings.

We revisit the construction of small cancellation quotients of amalgamated free products, and verify that it provides examples of hyperbolic TDLC-groups of rational discrete cohomological dimension $2$ when applied to amalgamated products of profinite groups over open subgroups.

We raise the question of whether our main result can be extended to locally compact hyperbolic groups if rational discrete cohomological dimension is replaced by asymptotic dimension. We prove that this is the case for discrete groups and sketch an argument for TDLC-groups.

In this paper, we study the asymptotic Plateau problem in ${ℍ}^2\times ℝ$. We construct the first examples of non-fillable finite curves with no thin tail in $S_{\infty }^1\times ℝ$.

Suppose a sequence $M_j$ of Alexandrov spaces collapses to a space $X$ with only weak singularities. Yamaguchi constructed a map $f_j:M_j\to X$ called an almost Lipschitz submersion for large $j$. We prove that if $M_j$ has a uniform positive lower bound for the volumes of spaces of directions, which is sufficiently large compared to the weakness of singularities of $X$, then $f_j$ is a locally trivial fibration. Moreover, we show some properties on the intrinsic metric and the volume of the fibers of $f_j$.