Motion Planning on One-Dimensional Peano Continua

Abstract

We study the Lusternik-Schnirelmann category and topological complexity of 1-dimensional spaces. We define both invariants as lengths of suitable closed filtrations, as opposed to a more common definition based on open covers. Our main results provide a precise description of $\mathbf{cat}(X)$ and $\mathbf{TC}(X)$ of a 1-dimensional Peano continuum $X$ in terms of the wildness rank of $X$. A surprising consequence is that $\mathbf{cat}(X)$ and $\mathbf{TC}(X)$ of a general 1-dimensional space $X$ can be arbitrarily high, which is in stark contrast with the analogous results for 1-dimensional CW-complexes.