Analog category and complexity
type: Item
instance of: scholarly article
title: Analog category and complexity
author: Ben Knudsen
author: Shmuel Weinberger
published in: SIAM Journal on Applied Algebra and Geometry
publication date: 2024-08-28T00:00:00Z
review text: In Section 12 of [NATO Sci. Ser. II, Math. Phys. Chem. 217, 185--230 (2006; Zbl 1089.68131)], \textit{M. Farber} gave a probabilistic interpretation of the topological complexity (TC) of a space by viewing elements of fiberwise joins of path spaces as convex combinations of paths whose coefficients are seen as probabilities.\N\NWith this interpretation in mind, the authors of the present article introduce the notion of analog (sequential) TC as well as analog Lusternik-Schnirelmann category as special cases of a more general notion of analog sectional category of a fibration \(f\colon X \to Y\). To define the latter, one first puts a suitable topology on the set \(\mathcal{P}(f)\) of those finitely supported probability measures on \(X\) whose supports lie in a single fiber of \(f\). Elements of \(\mathcal{P}(f)\) can be represented by formal convex combinations of elements of \(X\) and the space \(\mathcal{P}(f)\) fibers over \(Y\). For \(n \in \mathbb{N}\) one further considers the subspace \(\mathcal{P}_n(f)\subset \mathcal{P}(f)\) of those probability measures whose supports have cardinality at most \(n\). These constructions and their properties are explored in Sections 2 to 4 of the article.\N\NThe analog sectional category of \(f\), denoted by \(\mathrm{asecat}(f)\), is defined as the minimal value of \(n\in \mathbb{N}_0\), such that \(\mathcal{P}(f) \to Y\) admits a continuous section whose image lies in \(\mathcal{P}_{n+1}(f)\), which is carried out in Section 5 of the article. The analog category and analog (sequential) TCs of a space, denoted by \(\mathrm{acat}(X)\) and \(ATC_r(X)\), respectively, are then obtained in Section 6 by taking the analog sectional category of the path fibration whose sectional category equals the respective invariant. Here, in contrast to the above probabilistic interpretation of TC, one does not obtain a probability distribution among an ordered family of continuous motion planning algorithms, but among possible paths between …
reviewed by: Stephan Mescher
Mathematics Subject Classification ID: 55M30
Mathematics Subject Classification ID: 60B05
zbMATH DE Number: 7902822
zbMATH Keywords: topological complexity
zbMATH Keywords: LS category
zbMATH Keywords: probability
zbMATH Keywords: aspherical spaces
MaRDI profile type: Publication
cites work: Right-angled Artin groups, polyhedral products and the -generating function
cites work: Topological complexity of the Klein bottle
cites work: Bounds for higher topological complexity of real projective space implied by BP
cites work: Topological complexity of motion planning
cites work: On the topological complexity of aspherical spaces
cites work: Higher topological complexity of aspherical spaces
cites work: Q4435763
cites work: New lower bounds for the topological complexity of aspherical spaces
cites work: On category, in the sense of Lusternik-Schnirelmann
cites work: Symmetric Joins and Weighted Barycenters
cites work: The topological complexity of pure graph braid groups is stably maximal
cites work: On higher analogs of topological complexity
cites work: On the topology of algorithms. I
cites work: A convenient category of topological spaces
DOI: 10.1137/24M1635004
description: scientific article; zbMATH DE number 7902822