Topological complexity

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 Michal Farber in 2003.

Definition

Let X be a topological space and $PX=\{\gamma :[0,1]\,\to \,X\}$ be the space of all continuous pathes 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 $\{U_{i}\}_{i=1}^{k}$
for each $i=1,\ldots ,k$ , there exists a local section $s_{i}:\,U_{i}\to \,PX.$

Examples

The 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 $F(\mathbb {R} ^{m},n)$ $F(\mathbb {R} ^{m},n)$ is the configuration space of n distinct points in the Euclidean m-space, then
- $TC(F(\mathbb {R} ^{m},n))=2n-1$ for m odd
- $TC(F(\mathbb {R} ^{m},n))=2n-2$ for m even.
For the Klein bottle, the topological complexity is not knwon (July 2012).