“Topological complexity”的意思、由来-开放百科全书

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Topological complexity

释义

Definition
Examples
References

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{{elucidate|date=July 2012}}, 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 . The topological complexity is the minimal number k such that

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

Examples

The topological complexity: TC(X) = 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.^[1]

References

1. ^https://arxiv.org/pdf/1612.03133.pdf

{{cite news|author=Farber, M.|title=Topological complexity of motion planning|journal=Discrete & Computational Geometry|volume=29 |issue=2|pages= 211–221|year=2003}}
Armindo Costa: Topological Complexity of Configuration Spaces, Ph.D. Thesis, Durham University (2010), online

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1 : Topology

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