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

\{Ui\}

k
i=1
of

X x X

,

i=1,\ldots,k

, there exists a local section

si:Ui\toPX.

Examples

Sn

is 2 for n odd and 3 for n even. For example, in the case of the circle

S1

, 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.

F(\Rm,n)

is the configuration space of n distinct points in the Euclidean m-space, then

TC(F(\Rm,n))=\begin{cases}2n-1&for{\itmodd}\ 2n-2&for{\itmeven.}\end{cases}

References

External links

Notes and References

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