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 [math]\displaystyle{ PX=\{\gamma: [0,1]\,\to\,X\} }[/math] be the space of all continuous paths in X. Define the projection [math]\displaystyle{ \pi: PX\to\,X\times X }[/math] by [math]\displaystyle{ \pi(\gamma)=(\gamma(0), \gamma(1)) }[/math]. The topological complexity is the minimal number k such that

• there exists an open cover [math]\displaystyle{ \{U_i\}_{i=1}^k }[/math] of [math]\displaystyle{ X\times X }[/math],
• for each [math]\displaystyle{ i=1,\ldots,k }[/math], there exists a local section [math]\displaystyle{ s_i:\,U_i\to\, PX. }[/math]

Examples

• The topological complexity: TC(X) = 1 if and only if X is contractible.
• The topological complexity of the sphere [math]\displaystyle{ S^n }[/math] is 2 for n odd and 3 for n even. For example, in the case of the circle [math]\displaystyle{ S^1 }[/math], 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 [math]\displaystyle{ F(\R^m,n) }[/math] is the configuration space of n distinct points in the Euclidean m-space, then

[math]\displaystyle{ TC(F(\R^m,n))=\begin{cases} 2n-1 & \mathrm{for\,\, {\it m}\,\, odd} \\ 2n-2 & \mathrm{for\,\, {\it m}\,\, even.} \end{cases} }[/math]

• 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

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