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John Oprea

Publications and source records attributed to John Oprea.

At least 19 recordsLinked to original sources
arXiv2026

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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arXiv2025

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, $\pi_1$-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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arXiv2025

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, \`a 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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arXiv2023

Sequential topological complexity of aspherical spaces and sectional categories of subgroup inclusions

We generalize results from topological robotics on the topological complexity (TC) of aspherical spaces to sectional categories of fibrations inducing subgroup inclusions on the level of fundamental groups. In doing so, we establish new lower bounds on sequential TCs of aspherical spaces as well as the parametrized TC of epimorphisms. Moreover, we generalize the Costa-Farber canonical class for TC to classes for sequential TCs and explore their properties. We combine them with the results on sequential TCs of aspherical spaces to obtain results on spaces that are not necessarily aspherical.

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arXiv2022

Sequential parametrized topological complexity and related invariants

Parametrized motion planning algorithms \cite{CFW} have a high degree of universality and flexibility; they generate the motion of a robotic system under a variety of external conditions. The latter are viewed as parameters and constitute part of the input of the algorithm. The concept of sequential parametrized topological complexity ${\sf TC}_r[p:E\to B]$ is a measure of the complexity of such algorithms. It was studied in \cite{CFW, CFW2} for $r=2$ and in \cite{FP} for $r\ge 2$. In this paper we analyse the dependence of the complexity ${\sf TC}_r[p:E\to B]$ on an initial bundle with structure group $G$ and on its fibre $X$ viewed as a $G$-space. Our main results estimate ${\sf TC}_r[p:E\to B]$ in terms of certain invariants of the bundle and the action on the fibre. Moreover, we also obtain estimates depending on the base and the fibre. Finally, we develop a calculus of sectional categories featuring a new invariant ${\sf secat}_f[p:E\to B]$ which plays an important role in the study of sectional category of towers of fibrations.

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arXiv2020

Right-angled Artin groups, polyhedral products and the TC-generating function

For a graph $Γ$, let $K(H_Γ,1)$ denote the Eilenberg-Mac Lane space associated to the right-angled Artin (RAA) group $H_Γ$ defined by $Γ$. We use the relationship between the combinatorics of $Γ$ and the topological complexity of $K(H_Γ,1)$ to explain, and generalize to the higher TC realm, Dranishnikov's observation that the topological complexity of a covering space can be larger than that of the base space. In the process, for any positive integer $n$, we construct a graph $\mathcal{O}_n$ whose TC-generating function has polynomial numerator of degree $n$. Additionally, motivated by the fact that $K(H_Γ,1)$ can be realized as a polyhedral product, we study the LS category and topological complexity of more general polyhedral product spaces. In particular, we use the concept of a strong axial map in order to give an estimate, sharp in a number of cases, of the topological complexity of a polyhedral product whose factors are real projective spaces. Our estimate exhibits a mixed cat-TC phenomenon not present in the case of RAA groups.

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arXiv2020

The Digital Hopf Construction

Various concepts and constructions in homotopy theory have been defined in the digital setting. Although there have been several attempts at a definition of a fibration in the digital setting, robust examples of these digital fibrations are few and far between. In this paper, we develop a digital Hopf fibration within the category of tolerance spaces. By widening our category to that of tolerance spaces, we are able to give a construction of this digital Hopf fibration which mimics the smooth setting.

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arXiv2019

Morita Invariance of Equivariant Lusternik-Schnirelmann Category and Invariant Topological Complexity

We use the homotopy invariance of equivariant principal bundles to prove that the equivariant ${\mathcal A}$-category of Clapp and Puppe is invariant under Morita equivalence. As a corollary, we obtain that both the equivariant Lusternik-Schnirelmann category of a group action and the invariant topological complexity are invariant under Morita equivalence. This allows a definition of topological complexity for orbifolds.

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arXiv2019

A Fundamental Group for Digital Images

We define a fundamental group for digital images. Namely, we construct a functor from digital images to groups, which closely resembles the ordinary fundamental group from algebraic topology. Our construction differs in several basic ways from previously established versions of a fundamental group in the digital setting. Our development gives a prominent role to subdivision of digital images. We show that our fundamental group is preserved by subdivision.

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arXiv2019

Subdivision of Maps of Digital Images

With a view towards providing tools for analyzing and understanding digitized images, various notions from algebraic topology have been introduced into the setting of digital topology. In the ordinary topological setting, invariants such as the fundamental group are invariants of homotopy type. In the digital setting, however, the usual notion of homotopy leads to a very rigid invariance that does not correspond well with the topological notion of homotopy invariance. In this paper, we establish fundamental results about subdivision of maps of digital images with $1$- or $2$-dimensional domains. Our results lay the groundwork for showing that the digital fundamental group is an invariant of a much less rigid equivalence relation on digital images, that is more akin to the topological notion of homotopy invariance. Our results also lay the groundwork for defining other invariants of digital images in a way that makes them invariants of this less rigid equivalence.

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arXiv2019

Homotopy Theory in Digital Topology

Digital topology is part of the ongoing endeavour to understand and analyze digitized images. With a view to supporting this endeavour, many notions from algebraic topology have been introduced into the setting of digital topology. But some of the most basic notions from homotopy theory remain largely absent from the digital topology literature. We embark on a development of homotopy theory in digital topology, and define such fundamental notions as function spaces, path spaces, and cofibrations in this setting. We establish digital analogues of basic homotopy-theoretic properties such as the homotopy extension property for cofibrations, and the homotopy lifting property for certain evaluation maps that correspond to path fibrations in the topological setting. We indicate that some depth may be achieved by using these homotopy-theoretic notions to give a preliminary treatment of Lusternik-Schnirelmann category in the digital topology setting. This topic provides a connection between digital topology and critical points of functions on manifolds, as well as other topics from topological dynamics.

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arXiv2019

Higher topological complexity of aspherical spaces

In this article we study the higher topological complexity ${\sf TC}_r(X)$ in the case when $X$ is an aspherical space, $X=K(π, 1)$ and $r\ge 2$. We give a characterisation of ${\sf TC}_r(K(π, 1))$ in terms of classifying spaces for equivariant Bredon cohomology. Our recent paper \cite{FGLO}, joint with M. Grant and G. Lupton, treats the special case $r=2$. We also obtain in this paper useful lower bounds for ${\sf TC}_r(π)$ in terms of cohomological dimension of subgroups of $π\timesπ\times \dots\times π$ ($r$ times) with certain properties. As an illustration of the main technique we find the higher topological complexity of the Higman's groups. We also apply our method to obtain a lower bound for the higher topological complexity of the right angled Artin (RAA) groups, which, as was established in \cite{GGY} by a different method (in a more general situation), coincides with the precise value. We finish the paper by a discussion of the ${\sf TC}$-generating function $\sum_{r=1}^\infty {\sf TC}_{r+1}(X)x^r$ encoding the values of the higher topological complexity ${\sf TC}_r(X)$ for all values of $r$. We show that in many examples (including the case when $X=K(H, 1)$ with $H$ being a RAA group) the ${\sf TC}$-generating function is a rational function of the form $\frac{P(x)}{(1-x)^2}$ where $P(x)$ is an integer polynomial with $P(1)={\sf cat}(X)$.

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arXiv2018

Bredon cohomology and robot motion planning

In this paper we study the topological invariant ${\sf {TC}}(X)$ reflecting the complexity of algorithms for autonomous robot motion. Here, $X$ stands for the configuration space of a system and ${\sf {TC}}(X)$ is, roughly, the minimal number of continuous rules which are needed to construct a motion planning algorithm in $X$. We focus on the case when the space $X$ is aspherical; then the number ${\sf TC}(X)$ depends only on the fundamental group $π=π_1(X)$ and we denote it ${\sf TC}(π)$. We prove that ${\sf TC}(π)$ can be characterised as the smallest integer $k$ such that the canonical $π\timesπ$-equivariant map of classifying spaces $$E(π\timesπ) \to E_{\mathcal D}(π\timesπ)$$ can be equivariantly deformed into the $k$-dimensional skeleton of $E_{\mathcal D}(π\timesπ)$. The symbol $E(π\timesπ)$ denotes the classifying space for free actions and $E_{\mathcal D}(π\timesπ)$ denotes the classifying space for actions with isotropy in a certain family $\mathcal D$ of subgroups of $π\timesπ$. Using this result we show how one can estimate ${\sf TC}(π)$ in terms of the equivariant Bredon cohomology theory. We prove that ${\sf TC}(π) \le \max\{3, {\rm cd}_{\mathcal D}(π\timesπ)\},$ where ${\rm cd}_{\mathcal D}(π\timesπ)$ denotes the cohomological dimension of $π\timesπ$ with respect to the family of subgroups $\mathcal D$. We also introduce a Bredon cohomology refinement of the canonical class and prove its universality. Finally we show that for a large class of principal groups (which includes all torsion free hyperbolic groups as well as all torsion free nilpotent groups) the essential cohomology classes in the sense of Farber and Mescher are exactly the classes having Bredon cohomology extensions with respect to the family $\mathcal D$.

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arXiv2018

An upper bound for topological complexity

In arXiv:1711.10132 a new approximating invariant ${\mathsf{TC}}^{\mathcal{D}}$ for topological complexity was introduced called $\mathcal{D}$-topological complexity. In this paper, we explore more fully the properties of ${\mathsf{TC}}^{\mathcal{D}}$ and the connections between ${\mathsf{TC}}^{\mathcal{D}}$ and invariants of Lusternik-Schnirelmann type. We also introduce a new $\mathsf{TC}$-type invariant $\widetilde{\mathsf{TC}}$ that can be used to give an upper bound for $\mathsf{TC}$, $$\mathsf{TC}(X)\le {\mathsf{TC}}^{\mathcal{D}}(X) + \left\lceil \frac{2\dim X -k}{k+1}\right\rceil,$$ where $X$ is a finite dimensional simplicial complex with $k$-connected universal cover $\tilde X$. The above inequality is a refinement of an estimate given by Dranishnikov.

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arXiv2016

Hereditary properties of co-Kähler manifolds

We show how certain topological properties of co-K{ä}hler manifolds derive from those of the Kähler manifolds which construct them. We go beyond Betti number results and describe the cohomology algebra structure of co-Kähler manifolds. As a consequence, we prove that co-Kähler manifolds satisfy the Toral Rank Conjecture: $\dim(H^*(M;\mathbb{Q})) \geq 2^r$, for any $r$-torus $T^r$ which acts almost freely on $M$.

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arXiv2016

Parallel forms, co-Kähler Manifolds and their Models

We show how certain topological properties of co-Kähler manifolds derive from those of the Kähler manifolds which construct them. In particular, we show that the existence of parallel forms on a co-Kähler manifold reduces the computation of cohomology from the de Rham complex to certain amenable sub-cdga's defined by geometrically natural operators derived from the co-Kähler structure. This provides a simpler proof of the formality of the foliation minimal model in this context.

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arXiv2016

A splitting theorem for compact Vaisman manifolds

We extend to metric compact mapping tori a splitting result for coKähler manifolds. In particular, we prove that a compact Vaisman manifold is finitely covered by the product of a Sasakian manifold and a circle.

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