On sequential versions of distributional topological complexity

TL;DR

The paper introduces the sequential distributional topological complexity $\mathsf{dTC}_m(X)$, a non-decreasing family extending $\mathsf{dTC}(X)$ and paralleling the classical $\mathsf{TC}_m(X)$. It proves $\mathsf{dTC}_m$ is a homotopy invariant and relates it to distributional LS-category $\mathsf{dcat}$, establishing $\mathsf{dTC}_m(X)\le\mathsf{TC}_m(X)$ and $\mathsf{dcat}(X^{m-1})\le\mathsf{dTC}_m(X)\le\mathsf{dcat}(X^m)$, while showing these invariants can differ markedly yet coincide in key examples. Cohomological lower bounds are developed via symmetric products, yielding sharp results for spheres, surfaces, and products, and connecting $\mathsf{dTC}_m$ to distributional sectional category $\mathsf{dsecat}$ and distributional LS-category; the framework also yields exact values for several manifolds and highlights when $\mathsf{dTC}_m$ matches the classical $\mathsf{TC}_m$ (e.g., for products of spheres, orientable surfaces). The paper further explores analog invariants (acat, ATC_m) and establishes general inequalities linking distributional and analog approaches, offering a cohesive picture of sequential distributional complexity and its geometric and algebraic constraints.

Abstract

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.

Theorems & Definitions (77)

• Lemma 2.1
• proof
• Proposition 2.2: Ja
• Theorem 2.3
• Definition 3.1
• Definition 3.2
• Proposition 3.3
• proof
• Proposition 3.4
• Proposition 3.5