View source for 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{{elucidate|date=July 2012}}, introduced by Michael Farber in 2003.

==Definition==
Let ''X'' be a topological space and <math>PX=\{\gamma: [0,1]\,\to\,X\}</math> be the space of all continuous paths in ''X''. Define the projection <math>\pi: PX\to\,X\times X</math>
by <math>\pi(\gamma)=(\gamma(0), \gamma(1))</math>. The topological complexity is the minimal number ''k'' such that
*there exists an [[open cover]] <math>\{U_i\}_{i=1}^k</math> of <math>X\times X</math>,
*for each <math>i=1,\ldots,k</math>, there exists a [[Section (fiber bundle)|local section]] <math>s_i:\,U_i\to\, PX.</math>

==Examples==
*The topological complexity: TC(''X'')&nbsp;=&nbsp;1 if and only if ''X'' is [[contractible space|contractible]]. 
*The topological complexity of the [[n-sphere|sphere]] <math>S^n</math> is 2 for ''n'' odd and 3 for ''n'' even. For example, in the case of the [[circle]] <math>S^1</math>, we may define a path between two points to be the [[geodesics|geodesic]] between the points, if it is unique. Any pair of [[antipodal points]] can be connected by a counter-clockwise path.
*If <math>F(\R^m,n)</math> is the [[Configuration space (mathematics)|configuration space]] of ''n'' distinct points in the Euclidean ''m''-space, then
::<math>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.<ref>{{Cite arXiv |eprint = 1612.03133|last1 = Cohen|first1 = Daniel C.|title = Topological Complexity of the Klein bottle|last2 = Vandembroucq|first2 = Lucile|class = math.AT|year = 2016}}</ref>

==References==
{{Reflist}}
* {{cite news|author=Farber, M.|title=Topological complexity of motion planning|journal=Discrete & Computational Geometry|volume=29 |issue=2|pages= 211–221|year=2003}}
*Armindo Costa: ''Topological Complexity of Configuration Spaces'', Ph.D. Thesis, Durham University (2010), [http://etheses.dur.ac.uk/736/1/thesis__ArmindoCosta.pdf?DDD21+ online]

== External links ==

* [[nlab:topological+complexity|Topological complexity]] on [[nLab]]

[[Category:Topology]]

{{Topology-stub}}

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