Topological complexity

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In mathematics, topological complexity of a topological space X (also denoted by TC(X)) is a topological invariant closely connected to the motion planning problem[further explanation needed], introduced by Michal Farber in 2003.


Let X be a topological space and PX=\{\gamma: [0,1]\,\to\,X\} be the space of all continuous paths in X. Define the projection \pi: PX\to\,X\times X by \pi(\gamma)=(\gamma(0), \gamma(1)). The topological complexity is the minimal number k such that


  • The topological complexity: TC(X) = 1 if and only if X is contractible.
  • The topological complexity of the sphere S^n is 2 for n odd and 3 for n even. For example, in the case of the circle S^1, we may define a path between two points to be the geodesics, if it is unique. Any pair of antipodal points can be connected by a counter-clockwise path.
  • If F(\R^m,n) is the configuration space of n distinct points in the Euclidean m-space, then
TC(F(\R^m,n))=\begin{cases} 2n-1 & \mathrm{for\,\, {\it m}\,\, odd} \\ 2n-2 & \mathrm{for\,\, {\it m}\,\, even.} \end{cases}
  • For the Klein bottle, the topological complexity is not known (July 2012).


  • Farber, M. (2003). "Topological complexity of motion planning". Discrete & computational geometry. 29 (2). pp. 211–221. 
  • Armindo Costa: Topological Complexity of Configuration Spaces, Ph.D. Thesis, Durham University (2010), online