Lusternik-Schnirelmann category of the configuration space of complex projective space

TL;DR

This paper computes the Lusternik-Schnirelmann category and topological complexity of the two-point ordered configuration space of complex projective spaces, providing insights relevant to critical point theory and robotics motion planning.

Contribution

It provides the first explicit calculations of these invariants for the configuration space of complex projective spaces for all dimensions.

Findings

Calculated $cat$ and $TC$ for configuration spaces of $ ext{CP}^n$

Established relationships between $cat$ and $TC$ in this context

Extended known results to all $n \\geq 1$

Abstract

The Lusternik-Schnirelmann category $cat(X)$ is a homotopy invariant which is a numerical bound on the number of critical points of a smooth function on a manifold. Another similar invariant is the topological complexity $TC(X)$ (a la Farber) which has interesting applications in Robotics, specifically, in the robot motion planning problem. In this paper we calculate the Lusternik-Schnirelmann category and as a consequence we calculate the topological complexity of the two-point ordered configuration space of $\mathbb{CP}^n$ for every $n\geq 1$.