On sequential versions of distributional topological complexity

Abstract

We define a (non-decreasing) sequence ${\left\{{\mathrm{dTC}}_m\right(X\left)\right\}}_{m\ge 2}$ of sequential versions of distributional topological complexity ($\mathrm{dTC}$) of a space X introduced by Dranishnikov and Jauhari [5]. This sequence generalizes $\mathrm{dTC}\left(X\right)$ in the sense that ${\mathrm{dTC}}_2\left(X\right)=\mathrm{dTC}\left(X\right)$, and is a direct analog to the well-known sequence ${\left\{{\mathrm{TC}}_m\right(X\left)\right\}}_{m\ge 2}$. We show that like ${\mathrm{TC}}_m$ and $\mathrm{dTC}$, the sequential versions ${\mathrm{dTC}}_m$ are also homotopy invariants. Furthermore, ${\mathrm{dTC}}_m\left(X\right)$ relates with the distributional LS-category ($\mathrm{dcat}$) of products of X in the same way as ${\mathrm{TC}}_m\left(X\right)$ relates with the classical LS-category ($\mathrm{cat}$) of products of X. On one hand, we show that in general, ${\mathrm{dTC}}_m$ is a different concept than ${\mathrm{TC}}_m$ for each $m\ge 2$. On the other hand, by finding sharp cohomological lower bounds to ${\mathrm{dTC}}_m\left(X\right)$, we provide various examples of closed manifolds X for which the sequences ${\left\{{\mathrm{TC}}_m\right(X\left)\right\}}_{m\ge 2}$ and ${\left\{{\mathrm{dTC}}_m\right(X\left)\right\}}_{m\ge 2}$ coincide.