Arturo Espinosa Baro (3/28/24): Effective topological complexity and effective LS category
Title: On properties of effective topological complexity and effective Lusternik-Schnirelmann category
Abstract: Since the inception of the original notion of Topological Complexity, many different variants of the concept have been developed through the years, to capture distinct kinds of information that may be of interest for the motion planning problem. One of those kinds of ''specific information" concerns the impact of the symmetries that often appear in the configuration spaces. Formally, those symmetries are seen as actions of groups on the base topological space X and, as such, this naturally leads to the consideration of equivariant versions of topological complexity. There are several non-equivalent approaches to the matter. In this talk we will make a very brief review of them, but we will focus on the known as Effective Topological Complexity. This variant was devised by Zbigniew Błaszczyk and Marek Kaluba as a way to reduce the complexity of the motion planning problem through the symmetries of the configuration space by means of acknowledging the physically different but functionally equivalent states of the mechanical system that may appear, and which are linked by those symmetries. We will introduce a notion of Effective Lusternik-Schnirelmann category, and we will investigate some of the properties of both effective TC and cat, in particular their relationship with the orbit projection map of the group action and some non-vanishing conditions for effective TC at stage 2. This is a joint work with Zbigniew Błaszczyk and Antonio Viruel.