Overview
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, introduced by Michael Farber in 2003.
Definition
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 *there exists an open cover \ U_i\ _ i=1 ^k of X\times X , *for each i=1,\ldots,k , there exists a local section s_i:\,U_i\to\, PX.
Examples
*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 geodesic between the points, 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 *The topological complexity of the Klein bottle is 5.
References
* *Armindo Costa: Topological Complexity of Configuration Spaces, Ph.D. Thesis, Durham University (2010), online
External links
* Topological complexity on nLab ko:단면 범주#위상 복잡도