Consider the space X of all possible configurations of a mechanical system. A motion planning algorithm assigns to each pair of initial and final configurations (x,y) a continuous motion of the system starting at x and ending at y. Topological complexity is an integer TC(X) reflecting the complexity of motion planning algorithms for all systems having X as their configuration space. Roughly, TC(X) is the minimal number of continuous rules which are needed in a motion planning algorithm. This invariant was introduced by Farber in 2002 and is closely related to the classical Lusternik-Schnirelmann category.
In recent years, several versions of topological complexity aimed at exploiting the presence
of a group action in the configuration space have appeared. We will present several approaches to describing equivariant topological complexity variants and discuss possible interpretations.
Morita equivalence is a relationship between group actions that preserves many intrinsic properties of the action. We prove invariance under Morita equivalence for both the equivariant LS-category and the invariant topological complexity.
This is joint work with Andres Angel, Mark Grant and John Oprea. |