The Lusternik – Schnirelmann category (for short, LS-category) is a classical homotopy invariant dating back to Lusternik’s and Schnirelmann’s paper in 1929. Historically, their invariant gives a lower bound on the number of critical points for a smooth real valued function on a closed manifold. Since then, this numerical invariant has become a tool for many branches of mathematics. Of particular interest to me is a decade old question by Mark Grant that states that cohomological dimension of a group homomorphism coincides with its LS-category. The question was motivated by the Eilenberg-Ganea theorem from the 1950s saying that these two invariants agree for groups.

On the other hand, topological complexity (introduced by Farber) and higher topological complexity (introduced by Rudyak) are emerging areas of research pertaining to the study of topology in the context of applications to robotics. In particular, these fields study the topology of configuration spaces and the complexity of motion planning in such spaces. My research uses algebraic topology, particular LS-category and cohomological dimension, to explore topological complexity and sequential topological complexity of group homomorphisms.