Topological complexity explained

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 Michael Farber in 2003.

Definition

Let X be a topological space and

PX = \{\gamma: [ 0 , 1 ]\toX\} be the space of all continuous paths in X. Define the projection \pi: PX\toX x X by \pi(\gamma) = (\gamma( 0 ), \gamma( 1 )) . The topological complexity is the minimal number k such that\{U _{i\}} ^{k} ^{i =1} of X x X ,

• for each

i =1 ,\ldots,k , there exists a local section s _{i:U} _{i\to} PX.

Examples

• The topological complexity: TC(X) = 1 if and only if X is contractible.

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 geodesic between the points, 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, thenTC(F(\R ^{m,n))=\begin{cases}} 2 n -1 & for{\itm odd} \ 2 n -2 & for{\itm even.} \end{cases}

• The topological complexity of the Klein bottle is 5.^[1]

References

• News: Farber, M.. Topological complexity of motion planning. Discrete & Computational Geometry. 29 . 2. 211–221. 2003.
• Armindo Costa: Topological Complexity of Configuration Spaces, Ph.D. Thesis, Durham University (2010), online

External links

• Topological complexity on nLab

Notes and References

1. 1612.03133. Cohen. Daniel C.. Topological Complexity of the Klein bottle. Vandembroucq. Lucile. math.AT. 2016.