Motion Planning on One-Dimensional Peano Continua

TL;DR

This paper investigates the Lusternik-Schnirelmann category and topological complexity of 1-dimensional spaces, revealing that these invariants can be arbitrarily high for general spaces, unlike CW-complexes.

Contribution

It introduces a new approach to defining these invariants via closed filtrations and provides a precise characterization for 1-dimensional Peano continua.

Findings

$ extbf{cat}(X)$ and $ extbf{TC}(X)$ depend on the wildness rank of $X$

These invariants can be arbitrarily high for general 1D spaces

Contrast with fixed bounds for 1D CW-complexes

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.