Topological complexity, fibrations and symmetry

TL;DR

This paper explores how group actions and fibrations can be used to derive new upper bounds for the topological complexity of manifolds and groups, advancing understanding in algebraic topology.

Contribution

It introduces novel bounds for topological complexity using smooth group actions and applies these results to finitely generated torsion-free nilpotent groups.

Findings

New upper bounds for topological complexity of manifolds

Bounds for topological complexity of certain groups

Application of group actions to topological invariants

Abstract

We show how locally smooth actions of compact Lie groups on a manifold $X$ can be used to obtain new upper bounds for the topological complexity $\TC(X)$, in the sense of Farber. We also obtain new bounds for the topological complexity of finitely generated torsion-free nilpotent groups.