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
be the space of all continuous paths in
X. Define the projection
by
\pi(\gamma)=(\gamma(0),\gamma(1))
. The topological complexity is the minimal number
k such that
of
,
, there exists a
local section
Examples
- The topological complexity: TC(X) = 1 if and only if X is contractible.
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.
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
- 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
- 1612.03133. Cohen. Daniel C.. Topological Complexity of the Klein bottle. Vandembroucq. Lucile. math.AT. 2016.