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topological complexity (changes) in nLab

It is also possible to define it using the Schwarz genus? of the path space fibration ΩX↪PX→X\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

\begin{proposition} A topological space XX is contractible iff TC(X)=1TC(X)=1. \end{proposition}

(Farber 01, Theorem 1)

\begin{proposition} The topological complexity is only dependent on the homotopy type of a topological space. \end{proposition}

(Farber 01, Theorem 3)

Special topological complexities

Spheres and tori

\begin{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) \end{proposition}

(Farber 01, Theorem 8)

This theorem can be generalized:

\begin{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). \end{proposition}

(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

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

(Farber & Tabachnikov & Yuzvinsky 02, Theorem 12)

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

(Farber & Tabachnikov & Yuzvinsky 02, Proposition 18)

\begin{proposition} For any n∈ℕn\in\mathbb{N}, one has TC(ℂP n)=2n+1\operatorname{TC}(\mathbb{C}P^n)=2n+1 (with convention TC(*)=1\operatorname{TC}(*)=1). \end{proposition}

(Farber & Tabachnikov & Yuzvinsky 02, Corollary 2)

Σ\Sigma and Ξ\Xi surfaces

\begin{proposition} The topological complexity of a Σ\Sigma surface is

(3)TC(Σ g)={3 m≤1 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) \end{proposition}

(Farber 01, Theorem 9)

\begin{proposition} For n≥2n \geq 2 and g≥2g \geq 2 one has

(4)TC((ℝP 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). \end{proposition}

(Cohen & Vandembrouq 18, Theorem 1.3.)

Klein bottle

\begin{proposition} The topological complexity of the Klein bottle is 44 (with convention TC(*)=0TC(*)=0). \end{proposition}

(Cohen & Vandembrouq 16, Theorem 1)

Configuration space

\begin{proposition} The topological complexity of a configuration space is

(5)TC(Conf(ℝ m,n))={2n−1 modd 2n−2 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). \end{proposition}

(Farber & Grant 08, Theorem 1)