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*The topological complexity: TC(''X'') = 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]], 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>
*For the [[Klein bottle]], the topological complexity is not known (July 2012).
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