Higher topological complexity and its symmetrization

Understand

We develop the properties of the $n$-th sequential topological complexity $TC_n$, a homotopy invariant introduced by the third author as an extension of Farber's topological model for studying the complexity of motion planning algorithms in robotics.

• We exhibit close connections of $TC_n(X)$ to the Lusternik-Schnirelmann category of cartesian powers of $X$, to the cup-length of the diagonal embedding $X\hookrightarrow X^n$, and to the ratio between homotopy dimension and connectivity of $X$.
• We fully compute the numerical value of $TC_n$ for products of spheres, closed 1-connected symplectic manifolds, and quaternionic projective spaces.
• Our study includes two symmetrized versions of $TC_n(X)$.