Topological complexity

Definition

• there exists an open cover of,
• for each, there exists a local section

  Examples

• The topological complexity: TC = 1 if and only if X is contractible.
• The topological complexity of the sphere is 2 for n odd and 3 for n even. For example, in the case of the circle, we may define a path between two points to be the geodesic between the points, if it is unique. Any pair of antipodal points can be connected by a counter-clockwise path.
• If is the configuration space of n distinct points in the Euclidean m-space, then
• The topological complexity of the Klein bottle is 4.