# 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 Michal 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 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).

## References

• 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