Topological complexity: Difference between revisions

Revision as of 01:19, 14 July 2012

Revision as of 01:33, 14 July 2012

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*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]], 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]] of ''n'' distinct points in the Euclidean ''m''-space, then

**:: <math>TC(F(\R^m,n))=\begin{cases} 2n-1</math> & \text{for ''$m''$ odd} \\ 2n-2 & \text{for $m$ even}. \end{cases}</math>

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

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