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 $\mathrm{ MTC}_r(X)$.
Categories: math.AT