nLab topological complexity

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

The topological complexity [Farber 2001] of a topological space is a topological invariant related to the problem of motion planning and the immersion problem of real projective space.

Definition

For a topological space XX its topological complexity TC(X)TC(X) is the smallest number nn, so that there is an open cover (U k) k=1 n(U_k)_{k=1}^n of X×XX\times X by nn open subsets addmitting local sections s k:X×XPXs_k\colon X\times X\rightarrow P X of the evaluation map

PX X×X γ (γ(0),γ(1)) \array{ P X &\longrightarrow& X\times X \\ \gamma &\mapsto& \big(\gamma(0),\gamma(1)\big) }

of the path space PXP X.

It is also possible to define it using the Schwarz genus? of the path space fibration ΩXPXX\Omega X\hookrightarrow P X\rightarrow X.

There is also the convention of using the smallest number nn, so that an open cover (U k) k=0 n(U_k)_{k=0}^n of n+1n+1 open sets of X×XX\times X with the above property exists. This lowers all topological complexities by one, hence the convention used can be given by giving the topological complexity of the set with one point. (TC(*)=1TC(*)=1 for the upper convention and TC(*)=0TC(*)=0 for the lower convention.)

Properties

Proposition

A topological space XX is contractible iff TC(X)=1TC(X)=1.

(Farber 01, Theorem 1)

Proposition

The topological complexity is only dependent on the homotopy type of a topological space.

(Farber 01, Theorem 3)

Special topological complexities

Spheres and tori

Proposition

The topological complexity of a sphere is

(1)TC(S n)={2 nodd 3 neven TC \big( S^n \big) \;=\; \left\{ \begin{array}{ll} 2 & n \; odd \\ 3 & n \; even \end{array} \right.

(with convention TC(*)=1\operatorname{TC}(*)=1)

(Farber 01, Theorem 8)

This theorem can be generalized:

Proposition

The topological complexity of a product of spheres is

(2)TC((S m) n)={n+1 modd 2n+1 meven TC \big( (S^m)^n \big) \;=\; \left\{ \begin{array}{ll} n+1 & m \; odd \\ 2n+1 & m \; even \end{array} \right.

(with convention TC(*)=1\operatorname{TC}(*)=1).

(Farber 01, Theorem 13)

A special case of this proposition is TC(T n)=n+1TC(T^n)=n+1 for the topological complexity of the torus.

Real and complex projective space

Proposition

For n1,3,7n\neq 1,3,7, the smallest natural number kk\in\mathbb{N}, so that there exists an immersion of real projective space P n\mathbb{R}P^n into euclidean space k1\mathbb{R}^{k-1} is the topologial complexity TC(P n)\operatorname{TC}(\mathbb{R}P^n) (with convention TC(*)=1\operatorname{TC}(*)=1).

(Farber & Tabachnikov & Yuzvinsky 02, Theorem 12)

Proposition

For n=1,3,7n=1,3,7, one has TC(P n)=n+1\operatorname{TC}(\mathbb{R}P^n)=n+1 (with convention TC(*)=1\operatorname{TC}(*)=1).

(Farber & Tabachnikov & Yuzvinsky 02, Proposition 18)

Proposition

For any nn\in\mathbb{N}, one has TC(P n)=2n+1\operatorname{TC}(\mathbb{C}P^n)=2n+1 (with convention TC(*)=1\operatorname{TC}(*)=1).

(Farber & Tabachnikov & Yuzvinsky 02, Corollary 2)

Σ\Sigma and Ξ\Xi surfaces

Proposition

The topological complexity of a Σ\Sigma surface is

(3)TC(Σ g)={3 m1 5 m>1 TC \big( \Sigma_g \big) \;=\; \left\{ \begin{array}{ll} 3 & m \leq 1 \\ 5 & m \gt 1 \end{array} \right.

(with convention TC(*)=1TC(*)=1)

(Farber 01, Theorem 9)

Proposition

For n2n \geq 2 and g2g \geq 2 one has

(4)TC((P n) g)=2n TC \big( (\mathbb{R}P^n)^g \big) \;=\; 2n

for the connected sum of real projective space (with convention TC(*)=0TC(*)=0).

(Cohen & Vandembrouq 18, Theorem 1.3.)

Klein bottle

Proposition

The topological complexity of the Klein bottle is 44 (with convention TC(*)=0TC(*)=0).

(Cohen & Vandembrouq 16, Theorem 1)

Configuration space

Proposition

The topological complexity of a configuration space is

(5)TC(Conf( m,n))={2n1 modd 2n2 meven TC \big( Conf(\mathbb{R}^m,n) \big) \;=\; \left\{ \begin{array}{ll} 2n-1 & m \; odd \\ 2n-2 & m \; even \end{array} \right.

(with convention TC(*)=1TC(*)=1).

(Farber & Grant 08, Theorem 1)

References

Definition and basic properties of topological complexity:

See also:

On topological complexity of real projective space and connection with their immersion into cartesian space:

On topological complexity of connected sums:

On topological complexity of the Klein bottle:

On topological complexity of configuration space:

Last revised on February 14, 2024 at 07:49:21. See the history of this page for a list of all contributions to it.