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, introduced by Michal Farber in 2003.
Let X be a topological space and P X = { γ : [ 0 , 1 ] → X } be the space of all continuous paths in X. Define the projection π : P X → X × X by π ( γ ) = ( γ ( 0 ) , γ ( 1 ) ) . The topological complexity is the minimal number k such that
there exists an open cover { U i } i = 1 k of X × X ,for each i = 1 , … , k , there exists a local section s i : U i → P X . The topological complexity: TC(X) = 1 if and only if X is contractible.The topological complexity of the sphere S n is 2 for n odd and 3 for n even. For example, in the case of the circle S 1 , 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 F ( R m , n ) is the configuration space of n distinct points in the Euclidean m-space, thenFor the Klein bottle, the topological complexity is not known (July 2012).