Topological complexity

Let X be a topological space and ${\displaystyle PX=\{\gamma :[0,1]\,\to \,X\}}$ be the space of all continuous paths in X. Define the projection ${\displaystyle \pi :PX\to \,X\times X}$ by ${\displaystyle \pi (\gamma )=(\gamma (0),\gamma (1))}$. The topological complexity is the minimal number k such that

• there exists an open cover ${\displaystyle \{U_{i}\}_{i=1}^{k}}$ of ${\displaystyle X\times X}$,
• for each ${\displaystyle i=1,\ldots ,k}$, there exists a local section ${\displaystyle s_{i}:\,U_{i}\to \,PX.}$

• The topological complexity: TC(X) = 1 if and only if X is contractible.
• The topological complexity of the sphere ${\displaystyle S^{n}}$ is 2 for n odd and 3 for n even. For example, in the case of the circle ${\displaystyle S^{1}}$, 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 ${\displaystyle F(\mathbb {R} ^{m},n)}$ is the configuration space of n distinct points in the Euclidean m-space, then

${\displaystyle TC(F(\mathbb {R} ^{m},n))={\begin{cases}2n-1&\mathrm {for\,\,{\it {m}}\,\,odd} \\2n-2&\mathrm {for\,\,{\it {m}}\,\,even.} \end{cases}}}$

• The topological complexity of the Klein bottle is 5.^[1]