Abstract:

The $r$-th topological complexity of a space $X$, $\mathrm{TC}_r(X)$, is defined to be the least integer $n$ such that $X^r$ is covered by $n$ open sets, each of which has a local homotopy section of the diagonal map $X\to X^r$. Farber and Opera asked for which finite CW-complex $X$ the generating function $$\mathcal{F}(X) = \sum_{r\ge 1}\mathrm{TC}_{r+1}(X)x^r$$ is of the form $$\frac{P(x)}{(1-t)^2}$$ where $P(x)$ is a polynomial with $P(1)=\mathrm{cat}(X)$. I will talk about some results on this question.

This talk is based on joint work with Michael Farber, Don Stanley, and Atsushi Yamaguchi.

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