Topological complexity

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 Michael Farber in 2003.

Definition

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Let X be a topological space and   be the space of all continuous paths in X. Define the projection   by  . The topological complexity is the minimal number k such that

  • there exists an open cover   of  ,
  • for each  , there exists a local section  

Examples

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  • The topological complexity: TC(X) = 1 if and only if X is contractible.
  • The topological complexity of the sphere   is 2 for n odd and 3 for n even. For example, in the case of the circle  , we may define a path between two points to be the geodesic between the points, if it is unique. Any pair of antipodal points can be connected by a counter-clockwise path.
  • If   is the configuration space of n distinct points in the Euclidean m-space, then
 

References

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  1. ^ Cohen, Daniel C.; Vandembroucq, Lucile (2016). "Topological Complexity of the Klein bottle". arXiv:1612.03133 [math.AT].
  • Farber, M. (2003). "Topological complexity of motion planning". Discrete & Computational Geometry. Vol. 29, no. 2. pp. 211–221.
  • Armindo Costa: Topological Complexity of Configuration Spaces, Ph.D. Thesis, Durham University (2010), online
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