An upper bound for topological complexity

TL;DR

This paper investigates a new invariant called $ ext{TC}^ ext{D}$ for topological complexity, explores its properties and connections to Lusternik-Schnirelmann invariants, and introduces an upper bound for $ ext{TC}$ based on this invariant.

Contribution

It introduces a new $ ext{TC}$-type invariant $ ilde{ ext{TC}}$ and establishes an upper bound for $ ext{TC}$ using $ ext{TC}^ ext{D}$, refining previous estimates.

Findings

Defined and analyzed properties of $ ext{TC}^ ext{D}$.

Established a new upper bound for $ ext{TC}$ involving $ ext{TC}^ ext{D}$ and the dimension of $X$.

Connected $ ext{TC}^ ext{D}$ with Lusternik-Schnirelmann invariants.

Abstract

In arXiv:1711.10132 a new approximating invariant ${\mathsf{TC}}^{\mathcal{D}}$ for topological complexity was introduced called $\mathcal{D}$-topological complexity. In this paper, we explore more fully the properties of ${\mathsf{TC}}^{\mathcal{D}}$ and the connections between ${\mathsf{TC}}^{\mathcal{D}}$ and invariants of Lusternik-Schnirelmann type. We also introduce a new $\mathsf{TC}$-type invariant $\widetilde{\mathsf{TC}}$ that can be used to give an upper bound for $\mathsf{TC}$, $$\mathsf{TC}(X)\le {\mathsf{TC}}^{\mathcal{D}}(X) + \left\lceil \frac{2\dim X -k}{k+1}\right\rceil,$$ where $X$ is a finite dimensional simplicial complex with $k$-connected universal cover $\tilde X$. The above inequality is a refinement of an estimate given by Dranishnikov.