On sequential versions of distributional topological complexity

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.

arXiv comment29 pages. Changes made based on the referee report

References in corpus (3)

• Higher topological complexity and its symmetrization
• Topology of parametrised motion planning algorithms
• Distributional Topological Complexity and LS-category

Cited by in corpus (2)

• Distributional category of manifolds
• LS-category and sequential topological complexity of symmetric products