On the growth of topological complexity

TL;DR

This paper investigates the growth behavior of the $r$-th topological complexity of a space, introducing a lower bound and analyzing its asymptotic properties, revealing connections to the Lusternik–Schnirelmann category.

Contribution

It introduces a new lower bound for the topological complexity of rational spaces and analyzes its growth, providing insights into the asymptotic behavior of these invariants.

Findings

The generating function for $ ext{TC}_r(X)$ is rational, of the form $rac{P(x)}{(1-x)^2}$.

The asymptotic growth of $ ext{TC}_r(X)$ relates to the Lusternik–Schnirelmann category of $X$.

A new lower bound $ ext{MTC}_r(X)$ is proposed and its growth is estimated.

Abstract

Let $\mathrm{TC}_r(X)$ denote the $r$-th topological complexity of a space $X$. In many cases, the generating function $\sum_{r\ge 1}\mathrm{TC}_{r+1}(X)x^r$ is a rational function $\frac{P(x)}{(1-x)^2}$ where $P(x)$ is a polynomial with $P(1)=\mathrm{cat}(X)$, that is, the asymptotic growth of $\mathrm{TC}_r(X)$ with respect to $r$ is $\mathrm{cat}(X)$. In this paper, we introduce a lower bound $\mathrm{MTC}_r(X)$ of $\mathrm{TC}_r(X)$ for a rational space $X$, and estimate the growth of $\mathrm{MTC}_r(X)$.