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On Sequential Versions of Distributional Topological Complexity

Ekansh Jauhari
Jan 2024
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摘要原文
We define a (non-decreasing) sequence $\{\text{dTC}_m(X)\}_{m\ge 2}$ of higher versions of distributional topological complexity ($\text{dTC}$) of a space $X$ introduced by Dranishnikov and Jauhari. This sequence generalizes $\text{dTC}(X)$ in the sense that $\text{dTC}_2(X) = \text{dTC}(X)$, and is a direct analog to the classical sequence $\{\text{TC}_m(X)\}_{m\ge 2}$. We show that like $\text{TC}_m$ and $\text{dTC}$, the sequential versions $\text{dTC}_m$ are also homotopy invariants. Also, $\text{dTC}_m(X)$ relates with the distributional LS-category ($d\text{cat}$) of products of $X$ in the same way as $\text{TC}_m(X)$ relates with the classical LS-category ($\text{cat}$) of products of $X$. We show that in general, $\text{dTC}_m$ is a different concept than $\text{TC}_m$ for each $m \ge 2$, but we also provide various examples of spaces $X$ for which the sequences $\{\text{TC}_m(X)\}_{m\ge 2}$ and $\{\text{dTC}_m(X)\}_{m\ge 2}$ coincide.
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