Topological complexity

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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 Michael Farber in 2003.

Let X be a topological space and ${\displaystyle PX=\{\gamma$ 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.^[1]

References

External links

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