topological complexity

Let X be a topological space and $PX=\backslash \{\backslash gamma:\; [0,1]\backslash ,\backslash to\backslash ,X\backslash \}$ be the space of all continuous paths in X. Define the projection $\backslash pi:\; PX\backslash to\backslash ,X\backslash times\; X$

by $\backslash pi(\backslash gamma)=(\backslash gamma(0),\; \backslash gamma(1))$. The topological complexity is the minimal number k such that

• there exists an open cover $\backslash \{U\_i\backslash \}\_\{i=1\}^k$ of $X\backslash times\; X$,
• for each $i=1,\backslash ldots,k$, there exists a local section $s\_i:\backslash ,U\_i\backslash to\backslash ,\; PX.$

• The topological complexity: TC(X) = 1 if and only if X is contractible.
• The topological complexity of the sphere $S^n$ 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 $F(\backslash R^m,n)$ is the configuration space of n distinct points in the Euclidean m-space, then

::

• The topological complexity of the Klein bottle is 5.{{Cite journal |arxiv = 1612.03133|last1 = Cohen|first1 = Daniel C.|title = Topological complexity of the Klein bottle|last2 = Vandembroucq|first2 = Lucile| journal=Journal of Applied and Computational Topology |year = 2016| volume=1 | issue=2 | pages=199–213 | doi=10.1007/s41468-017-0002-0 }}

{{Reflist}}

• {{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), [http://etheses.dur.ac.uk/736/1/thesis__ArmindoCosta.pdf?DDD21+ online]

• Topological complexity on nLab

Category:Topology

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