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, 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.