Topological complexity

Short description: Concept in topology

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 $i=1,\dots ,k$, 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

$TC(F({\mathbb{ℝ}}^m,n))=\{\begin{array}{ll}2n-1& \mathrm{f}\mathrm{o}\mathrm{r}{m}\mathrm{o}\mathrm{d}\mathrm{d}\\ 2n-2& \mathrm{f}\mathrm{o}\mathrm{r}{m}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}\mathrm{.}\end{array}$

• The topological complexity of the Klein bottle is 5.

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

External links

• Topological complexity on nLab

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