Topological complexity of configuration spaces

Farber, Michael; Grant, Mark

doi:10.1090/S0002-9939-08-09808-0

Topological complexity of configuration spaces
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by Michael Farber and Mark Grant PDF

Proc. Amer. Math. Soc. 137 (2009), 1841-1847 Request permission

Abstract:

The topological complexity $\mathsf {TC}(X)$ is a homotopy invariant which reflects the complexity of the problem of constructing a motion planning algorithm in the space $X$, viewed as configuration space of a mechanical system. In this paper we complete the computation of the topological complexity of the configuration space of $n$ distinct points in Euclidean $m$-space for all $m\ge 2$ and $n\ge 2$; the answer was previously known in the cases $m=2$ and $m$ odd. We also give several useful general results concerning sharpness of upper bounds for the topological complexity.

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