In mathematics, topological complexity of a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...

''X'' (also denoted by TC(''X'')) is a

topological invariant In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological spac ...

closely connected to the

motion planning Motion planning, also path planning (also known as the navigation problem or the piano mover's problem) is a computational problem to find a sequence of valid configurations that moves the object from the source to destination. The term is used ...

problem, introduced by Michael Farber in 2003.

Definition

Let ''X'' be a topological space and

PX=\

be the space of all continuous paths in ''X''. Define the projection

\pi: PX\to\,X\times X

\pi(\gamma)=(\gamma(0), \gamma(1))

. The topological complexity is the minimal number ''k'' such that *there exists an

open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alph ...

\_^k

X\times 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 In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within th ...

. *The topological complexity of the

sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

S^n

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

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

S^1

, we may define a path between two points to be the

geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...

between the points, if it is unique. Any pair of

antipodal points In mathematics, antipodal points of a sphere are those diametrically opposite to each other (the specific qualities of such a definition are that a line drawn from the one to the other passes through the center of the sphere so forms a true ...

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, then ::

TC(F(\R^m,n))=\begin 2n-1 & \mathrm \\ 2n-2 & \mathrm \end

*The topological complexity of the

Klein bottle In topology, a branch of mathematics, the Klein bottle () is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a ...

is 5.

References

* *Armindo Costa: ''Topological Complexity of Configuration Spaces'', Ph.D. Thesis, Durham University (2010)
online
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