Topological complexity: Difference between revisions

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{{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 MichalMichael 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</math> & \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 journal |arxiv = 1612.03133|last1 = Cohen|first1 = Daniel C.|title = Topological complexity of the Klein bottle|last2 = Vandembroucq|first2 = Lucile| journal=Journal of Applied and Computational Topology |year = 2016| volume=1 | issue=2 | pages=199–213 | doi=10.1007/s41468-017-0002-0 }}</ref>

**<math>TC(F(\R^m,n))=2n-2</math> for ''m'' even.

*For the [[Klein bottle]], the topological complexity is not known (July 2012).

==References==

{{Reflist}}

* {{cite articlenews|author=Farber, M.|title=Topological complexity of motion planning|journal=Discrete & computationalComputational geometryGeometry|volume=29 (|issue=2)|pages= 211-221211–221|year=19972003}}

*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]]

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