On totally c-symplectically aspherical manifolds
We construct examples of totally c-symplectically aspherical near-symplectic manifolds with non-trivial second homotopy group.
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arXiv subjects
Ekansh Jauhari
Publications and source records attributed to Ekansh Jauhari.
Symplectically aspherical Kähler manifolds, scalar curvature, and the fundamental group
We present a detailed study of closed smooth manifolds having Kähler forms that pullback to exact forms on the universal cover. We show that these manifolds, which we call symplectically aspherical Kähler manifolds, exist in abundance, even outside the aspherical setting, and have interesting topological and geometric features, such as large fundamental group á la Kollár and the absence of Kähler metrics of positive scalar curvature. Motivated by the latter, we extend the Gromov--Lawson Conjecture on aspherical manifolds to symplectically aspherical manifolds and prove it in the spin case. We also study Kähler cones on symplectically aspherical Kähler manifolds and the realizability problem of their fundamental group, and explore their other complex geometric properties.
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Curvature, macroscopic dimensions, and symmetric products of surfaces
We present a detailed study of the curvature and symplectic asphericity properties of symmetric products of surfaces. We show that these spaces can be used to answer nuanced questions arising in the study of closed Riemannian manifolds with positive scalar curvature. For example, we prove that symmetric products of surfaces sharply distinguish between two distinct notions of macroscopic dimension introduced by Gromov and the second-named author. As a natural generalization of this circle of ideas, we address the Gromov--Lawson and Gromov conjectures in the Kaehler projective setting and draw new connections between the theories of the minimal model, positivity in algebraic geometry, and macroscopic dimensions.
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Homotopic distances and group-like spaces
Homotopic distance is a numerical homotopy invariant that quantifies the homotopic distinction between two or more continuous maps. In this paper, we look at various versions of homotopic distance between maps and study them on maps to group-like spaces and CW H-spaces. In particular, (1) we develop the theories of probabilistic and diagonalized versions of the homotopic distance and compare their properties, and (2) we show that all versions of the homotopic distance can be calculated precisely in terms of their respective versions of the Lusternik-Schnirelmann category when group-like spaces or CW H-spaces are involved. Our results are refined in the setting of rational groups. On the way, we also study loopings of maps and the nilpotency of the set of homotopy classes of maps to group-like spaces.
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On the complexity of parametrized motion planning algorithms
We study a probabilistic variant of the r-th sequential parametrized topological complexity, which bounds this classical invariant from below and measures the difficulty in constructing permissive parametrized motion planning algorithms. On one hand, we use cohomology to show that this new invariant behaves similarly to the classical invariant on Fadell-Neuwirth fibrations and oriented sphere bundles; on the other hand, we use equivariant homotopy theory to prove that its behavior is wildly different on bundles whose fibers are real projective spaces and whose structure groups are special orthogonal groups. We also explore several other features of our invariant and its relationships with various other invariants motivated by topological robotics.
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Homotopy connectivity of Čech complexes of spheres
Let $S^n$ be the $n$-sphere with the geodesic metric and of diameter $π$. The intrinsic Čech complex of $S^n$ at scale $r$ is the nerve of all open balls of radius $r$ in $S^n$. In this paper, we show how to control the homotopy connectivity of Čech complexes of spheres at each scale between $0$ and $π$ in terms of coverings of spheres. Our upper bound on the connectivity, which is sharp in the case $n=1$, comes from the chromatic numbers of Borsuk graphs of spheres. Our lower bound is obtained using the conicity (in the sense of Barmak) of Čech complexes of the sufficiently dense, finite subsets of $S^n$. Our bounds imply the new result that for $n\ge 1$, the homotopy type of the Čech complex of $S^n$ at scale $r$ changes infinitely many times as $r$ varies over $(0,π)$; we conjecture only countably many times. Additionally, we lower bound the homological dimension of Čech complexes of finite subsets of $S^n$ in terms of their packings.
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On the sequential topological complexity and the LS-category of the cofiber of higher diagonals for symmetric products of non-orientable surfaces
For positive integers $k$, $n$, and $g$ with $k\geq2$, we give a closed-form expression for the $k$-th $\mathbb{Z}_2$-zero-divisor cup length $\mathsf{zcl}_k(SP^n(N_g))$ of the $n$-th symmetric product $SP^n(N_g)$ of the closed non-orientable surface $N_g$ of genus $g$. This allows us to estimate, and in some cases, completely determine, the $k$-th sequential topological complexity $\mathsf{TC}_k(SP^n(N_g))$, as well as the Lusternik--Schnirelmann category of the homotopy cofiber of the $k$-th diagonal map $SP^n(N_g) \to (SP^n(N_g))^k$. Our results recover previously known facts for even-dimensional real projective spaces ($g=1$) and closed non-orientable surfaces ($n=1$). In addition, we show that, as $g$ grows, $\mathsf{TC}_2(SP^n(N_g))$ behaves in a different way as all other invariants $\mathsf{TC}_k(SP^n(N_g))$ do. Likewise, as $k$ grows, we describe an eventual maximal-possible linear growth of $\mathsf{zcl}_k(SP^n(N_g))$, which allows us to prove the rationality conjecture of Farber and Oprea for the TC-generating function of $SP^n(N_g)$.
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Bochner-type theorems for distributional category
We show that in the presence of a geometric condition such as non-negative Ricci curvature, the distributional category of a manifold may be used to bound invariants, such as the first Betti number and macroscopic dimension, from above. Moreover, à la Bochner, when the bound is an equality, special constraints are imposed on the manifold. We show that the distributional category of a space also bounds the rank of the Gottlieb group, with equality imposing constraints on the fundamental group. These bounds are refined in the setting of cohomologically symplectic manifolds, enabling us to get specific computations for the distributional category and LS-category.
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On distributional one-category, diagonal distributional complexity, and related invariants
We develop the theory of probabilistic variants of the one-category and diagonal topological complexity, which bound the classical LS-category and topological complexity from below. Unlike any other classical or probabilistic invariants, these invariants are rigid on spaces with finite fundamental group. On Eilenberg-Mac Lane spaces, we identify these new invariants with distributional category and complexity, respectively, and use them to illuminate aspects of the behavior of the latter invariants on aspherical spaces and products of spaces. We also study their properties on covering maps, $π_1$-isomorphisms, $H$-spaces, and closed essential manifolds, and consequently, obtain the first examples of closed manifolds beyond the real projective spaces on which the distributional theory disagrees with the classical one.
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LS-category and sequential topological complexity of symmetric products
The $n$-th symmetric product of a topological space $X$ is the orbit space of the natural action of the symmetric group $S_n$ on the product space $X^n$. In this paper, we compute the sequential topological complexities of (finite products of) the symmetric products of closed orientable surfaces, thereby verifying the rationality conjecture of Farber and Oprea for these spaces. Additionally, we determine the Lusternik-Schnirelmann category of (finite products of) the symmetric products of closed non-orientable surfaces. More generally, we provide lower bounds to the LS-category and the sequential topological complexities of the symmetric products of finite CW complexes $X$ in terms of the cohomology of $X$ and its products. On the way, we also obtain new lower bounds to the sequential distributional complexities of continuous maps and study the homotopy groups of the symmetric products of closed surfaces.
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Distributional category of manifolds
Recently, a new homotopy invariant of metric spaces, called the distributional LS-category, was defined, which provides a lower bound to the classical LS-category. In this paper, we obtain several sufficient conditions for the distributional LS-category (dcat) of a closed manifold to be maximum, i.e., equal to its classical LS-category (cat). These give us many new computations of dcat, especially for some essential manifolds and (generalized) connected sums. In the process, we also determine the cat of closed 3-manifolds having torsion-free fundamental groups and some closed geometrically decomposable 4-manifolds. Finally, we extend some of our results to closed Alexandrov spaces with curvature bounded below and discuss their cat and dcat in dimension 3.
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On sequential versions of distributional topological complexity
We define a (non-decreasing) sequence $\{\mathsf{dTC}_m(X)\}_{m\ge 2}$ of higher versions of distributional topological complexity ($\mathsf{dTC}$) of a space $X$ introduced by Dranishnikov and Jauhari. This sequence generalizes $\mathsf{dTC}(X)$ in the sense that $\mathsf{dTC}_2(X) = \mathsf{dTC}(X)$, and is a direct analog to the classical sequence $\{\mathsf{TC}_m(X)\}_{m\ge 2}$. We show that like $\mathsf{TC}_m$ and $\mathsf{dTC}$, the sequential versions $\mathsf{dTC}_m$ are also homotopy invariants. Also, $\mathsf{dTC}_m(X)$ relates with the distributional LS-category ($\mathsf{dcat}$) of products of $X$ in the same way as $\mathsf{TC}_m(X)$ relates with the classical LS-category ($\mathsf{cat}$) of products of $X$. On one hand, we show that in general, $\mathsf{dTC}_m$ is a different concept than $\mathsf{TC}_m$ for each $m \ge 2$. On the other hand, by finding sharp cohomological lower bounds to $\mathsf{dTC}_m(X)$, we provide various examples of closed manifolds $X$ for which the sequences $\{\mathsf{TC}_m(X)\}_{m\ge 2}$ and $\{\mathsf{dTC}_m(X)\}_{m\ge 2}$ coincide.
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Intertwining category and complexity
We develop the theory of the intertwining distributional versions of the LS-category and the sequential topological complexities of a space $X$, denoted by $\mathsf{icat}(X)$ and $\mathsf{iTC}_m(X)$, respectively. We prove that they satisfy most of the nice properties as their respective distributional counterparts $\mathsf{dcat}(X)$ and $\mathsf{dTC}_m(X)$, and their classical counterparts $\mathsf{cat}(X)$ and $\mathsf{TC}_m(X)$, such as homotopy invariance and special behavior on topological groups. We show that the notions of $\mathsf{iTC}_m$ and $\mathsf{dTC}_m$ are different for each $m \ge 2$ by proving that $\mathsf{iTC}_m(\mathcal{H})=1$ for all $m \ge 2$ for Higman's group $\mathcal{H}$. Using cohomological lower bounds, we also provide various examples of locally finite CW complexes $X$ for which $\mathsf{icat}(X) > 1$, $\mathsf{iTC}_m(X) > 1$, $\mathsf{icat}(X) = \mathsf{dcat}(X) = \mathsf{cat}(X)$, and $\mathsf{iTC}(X) = \mathsf{dTC}(X) = \mathsf{TC}(X)$.
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Distributional Topological Complexity and LS-category
We define a new version of Topological Complexity (TC) of a space, denoted as $\text{dTC}$, which, we think, fits better for motion planning for some autonomous systems. Like Topological complexity, \text{dTC} is also a homotopy invariant. Also, $\text{dTC}$ has a corresponding analog, denoted as $\text{dcat}$, to the Lusternik-Schnirelmann category (cat). In this paper, we do computations and estimates for both $\text{dTC(X)}$ and $\text{dcat(X)}$ for some spaces $X$ as well as a comparison with $\text{TC(X)}$ and $\text{cat(X)}$.
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