On the growth of topological complexity

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 ${\mathrm{MTC}}_r(X)$.

An upper bound for the growth of TC in the first version is false and removed. Accepted by J. Appl. Compt. Topol

References in corpus (1)

• Joins of DGA modules and sectional category