Analog category and complexity (Q6594422)

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 the considered points without having to consider orderings among them. The authors show among other basic results that for a paracompact \(X\), it holds for each \(r \geq 2\) that\N\[\NATC_r(X) \leq TC_r(X).\N\]\NThe authors further demonstrate the difference between analog TC and TC by showing in Proposition 6.9 that \(ATC_2(\mathbb{R}P^n)=1\) for each \(n \in \mathbb{N}\), which is in stark contrast with the computations of \(TC(\mathbb{R}P^n)\) obtained by \textit{M. Farber} et al. [Int. Math. Res. Not. 2003, No. 34, 1853--1870 (2003; Zbl 1030.68089)].\N\NIn Section 7 the authors further study aspherical spaces and obtain an Eilenberg-Ganea-type theorem for analog category in Theorem 7.4, which states that for any torsion-free group \(G\) it holds that\N\[\N\mathrm{acat}(BG) = \mathrm{cd}(G),\N\]\Nthe cohomological dimension of \(G\). The following Section 8 contains the technicalities of the proof of a result from Section 2 while a concluding appendix recalls properties of convenient spaces, i.e. compactly generated Hausdorff spaces, which are used in large parts of the article.\N\NIt is worth pointing out that \textit{A. Dranishnikov} and \textit{E. Jauhari} have independently introduced the closely related notions of distributional topological complexity and distributional LS category in [EMS Ser. Ind. Appl. Math. 4, 363--385 (2024; Zbl 07947563)], which are defined in a very similar way as the analog invariants, but with respect to a different topology on the considered space of probability measures. The authors of the present article conjecture in its introduction that analog and distributional topological complexity coincide for metrizable spaces.