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[further explanation needed], introduced by Michael Farber in 2003.
Definition
Let X be a topological space and
be the space of all continuous paths in X. Define the projection
by
. The topological complexity is the minimal number k such that
- there exists an open cover
of
, - for each
, there exists a local section ![{\displaystyle s_{i}:\,U_{i}\to \,PX.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Examples
- The topological complexity: TC(X) = 1 if and only if X is contractible.
- The topological complexity of the sphere
is 2 for n odd and 3 for n even. For example, in the case of the circle
, 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
is the configuration space of n distinct points in the Euclidean m-space, then
![{\displaystyle TC(F(\mathbb {R} ^{m},n))={\begin{cases}2n-1&\mathrm {for\,\,{\it {m}}\,\,odd} \\2n-2&\mathrm {for\,\,{\it {m}}\,\,even.} \end{cases}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
References
- ^ Cohen, Daniel C.; Vandembroucq, Lucile (2016). "Topological Complexity of the Klein bottle". arXiv:1612.03133 [math.AT].
- 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