nLab topological complexity

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 $X$ its topological complexity $TC(X)$ is the smallest number $n$, so that there is an open cover $(U_k)_{k=1}^n$ of $X\times X$ by $n$ open subsets addmitting local sections $s_k\colon X\times X\rightarrow P X$ of the evaluation map

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

of the path space $P X$.

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

There is also the convention of using the smallest number $n$, so that an open cover $(U_k)_{k=0}^n$ of $n+1$ open sets of $X\times X$ 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. ($TC(*)=1$ for the upper convention and $TC(*)=0$ for the lower convention.)

Properties

Proposition

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

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

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

(Farber 01, Theorem 8)

This theorem can be generalized:

Proposition

The topological complexity of a product of spheres is

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

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

(Farber 01, Theorem 13)

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

Real and complex projective space

Proposition

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

(Farber & Tabachnikov & Yuzvinsky 02, Theorem 12)

Proposition

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

(Farber & Tabachnikov & Yuzvinsky 02, Proposition 18)

Proposition

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

(Farber & Tabachnikov & Yuzvinsky 02, Corollary 2)

$\Sigma$ and $\Xi$ surfaces

Proposition

The topological complexity of a $\Sigma$ surface is

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

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

(Farber 01, Theorem 9)

Proposition

For $n \geq 2$ and $g \geq 2$ one has

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

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

(Cohen & Vandembrouq 18, Theorem 1.3.)

Klein bottle

Proposition

The topological complexity of the Klein bottle is $4$ (with convention $TC(*)=0$).

(Cohen & Vandembrouq 16, Theorem 1)

Configuration space

Proposition

The topological complexity of a configuration space is

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

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

(Farber & Grant 08, Theorem 1)

References

Definition and basic properties of topological complexity:

• Michael Farber, Topological complexity of motion planning, Discrete Comput Geom 29 (2001) 211–221, [arXiv:math/0111197, doi:10.1007/s00454-002-0760-9]

See also:

• Wikipedia, Topological complexity

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

• Michael Farber, Serge Tabachnikov, Sergey Yuzvinsky, Topological robotics: motion planning in projective spaces (2002), arXiv:math/0210018;

On topological complexity of connected sums:

• Daniel C. Cohen, Lucile Vandembroucq, Motion planning in connected sums of real projective spaces (2018), arXiv:1807.09947

On topological complexity of the Klein bottle:

• Daniel C. Cohen, Lucile Vandembroucq, Topological Complexity of the Klein bottle (2018), arXiv:1612.03133v3

On topological complexity of configuration space:

• Michael Farber, Mark Grant, Topological Complexity of the Klein bottle (2008), arXiv:0806.4111

See also:

• Kenji Fukushi: Topological complexity for closed 1-forms [arXiv:2606.09184]

On topological complexity of symplectic 4-manifolds:

• Ryuma Orita: Maximal topological complexity of monotone symplectic 4-manifolds [arXiv:2607.20886]

On topological complexity of maps:

• Sutirtha Datta, Navnath Daundkar, Abhishek Sarkar, Soumen Sarkar: Discrete version of topological complexity of maps [arXiv:2607.21330]