Algebraic characterization of classical group sequential topological complexity

Determine an explicit algebraic description of the group invariants TC_r(BG) for all integers r ≥ 1 and for arbitrary discrete groups G, where TC_r(BG) denotes the rth sequential topological complexity of the classifying space BG; specifically, characterize these invariants in group-theoretic terms rather than via topological formulations, to parallel the sought algebraic description of the analog invariants ATC_r(BG).

Background

The authors propose describing the analog group invariants ATC_r(BG) algebraically and note that, for the classical "digital" invariants TC_r(BG), this kind of algebraic characterization is notoriously difficult. They emphasize that obtaining algebraic descriptions for the analog invariants could shed light on the classical case because the analog invariants provide lower bounds for the digital ones.

This open problem is central in the paper of sequential topological complexity, where even computations for specific groups are often challenging. An algebraic characterization of TC_r(BG) would significantly advance understanding of motion planning complexity in group-theoretic terms.