Motion Planning on One-Dimensional Peano Continua

TL;DR

We address the problem of computing Lusternik-Schnirelmann category and topological complexity for 1-dimensional spaces, including fractal-like Peano continua, where open covers are inadequate. The authors redefine these invariants using closed filtrations and introduce the wild set $\\operatorname{w}(X)$ and the wildness rank $\\mathbf{wrk}(X)$, analyzing iterated wild sets for $w$-stable spaces. For finite $\\mathbf{wrk}(X)=n$, they obtain exact or near-exact values: $\\mathbf{cat}(X)=n-1$ or $n$ and $\\mathbf{TC}(X)=2n-2$, $2n-1$, or $2n$ depending on the presence of simple closed curves in $\\operatorname{w}^{n-1}(X)$; if $\\mathbf{wrk}(X)=\\infty$, both invariants are infinite. The results reveal that 1D spaces beyond CW complexes can have arbitrarily large motion-planning complexity and provide a decomposition-based framework (deforestation) to bound and compute $\\mathbf{TC}$ via the wild-set hierarchy and associated retracts.

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.

Figures (4)

• Figure 1: A space whose four corners are wild points (left) and the Sierpinski Carpet (right), in which all points are wild .
• Figure 2: A Peano Continuum $X$ whose wild set $\operatorname{w}(X)$ is the outer boundary square (left) and a subspace $Y$ as in Theorem \ref{['structuretheorem']} that deformation retracts onto $\operatorname{w}(X)$ and for which $X\backslash Y$ is a disjoint union of open arcs (right). In the more general scenario, $X$ may have a null-sequence of dendrites $E_1,E_2,\dots$ attached to it. Whenever such a dendrite $E_n$ meets $X$ at a point of the subspace $Y$ shown here, we would include it to become part of $Y$.
• Figure 3: A Peano Continuum $X$ whose wild set $\operatorname{w}(X$) is the ternary Cantor set. Theorem \ref{['dendritetheorem']} applies to $X$ since there is an arc which contains $\operatorname{w}(X)$.
• Figure 4: A one-dimensional Peano continuum $X$ (left) whose wild set $\operatorname{w}(X)$ (right) is homotopy equivalent to the infinite earring space and thus $\mathbf{wrk}(X)=3$.

Theorems & Definitions (72)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Theorem 2.4
• proof
• Example 2.5
• Definition 2.6
• Proposition 2.7