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 ${\displaystyle PX=\{\gamma$ :[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 $F(\mathbb {R} ^{m},n)$ is the configuration space of n distinct points in the Euclidean m-space, then

$TC(F(\mathbb {R} ^{m},n))={\begin{cases}2n-1&\mathrm {for\,\,{\it {m}}\,\,odd} \\2n-2&\mathrm {for\,\,{\it {m}}\,\,even.} \end{cases}}$

• The topological complexity of the Klein bottle is 5.

References

• 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

External links

• Topological complexity on nLab