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(Sequential) topological complexity of aspherical spaces and sectional categories of subgroup inclusions
- Speaker(s)
- Arturo Espinosa Baro
- Affiliation
- UAM
- Language of the talk
- English
- Date
- Oct. 16, 2024, 10:30 a.m.
- Room
- room 4070
- Seminar
- Seminar Algebraic Topology
The topological complexity (TC) of a topological space is a homotopy invariant introduced by M. Farber to study the order of instability of motion planning algorithms of configuration spaces of mechanical and autonomous systems. Different variants of the original notion have emerged to answer different aspects of the motion planning problem. One of the most natural generalizations are the sequential topological complexities (TC_r for r \geq 2), developed by Y. Rudyak to model the motion planning problem for robots that are supposed to make some pre-determined intermediate stops along their ways.
One of the most important open problems on the field is the characterization on purely algebraic terms of the (sequential) topological complexity of aspherical spaces. This is motivated by the classic result of S. Eilenberg and T. Ganea stating that the Lusternik-Schnirelmann category of the classifying space of a torsion-free group corresponds with the cohomological dimension of the group. One of the possible approaches to the problem is through the study of the sectional categories of subgroup inclusions, as natural generalizations of sequential TCs for this algebraic setting.
We will make a brief overwiew of the recent developments on the problem, and then we will proceed to introduce how to obtain new lower bounds for sectional categories of subgroup inclusions through homological algebra methods, and to discuss their consequences for sequential topological complexity of aspherical spaces. If time permits, we will also mention how some of our methods allow to obtain results on spaces that are not necessarily aspherical.
This is mostly based on a joint work with Michael Farber, Stephan Mescher and John Oprea.
