Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Geometry of collapsing and free deformation retraction

Alexey Gorelov

Abstract

We show that a compact polyhedron $P$ collapses to a subpolyhedron $Q$ if and only if it admits a piecewise-linear free deformation retraction onto $Q$. We also consider further possibilities for invariant characterisations of collapsibility in terms of metrics; in this connection, we provide a partial correction to Isbell's claim that every injective metric space is freely contractible, and present a counterexample to a step in the original argument.

Geometry of collapsing and free deformation retraction

Abstract

We show that a compact polyhedron collapses to a subpolyhedron if and only if it admits a piecewise-linear free deformation retraction onto . We also consider further possibilities for invariant characterisations of collapsibility in terms of metrics; in this connection, we provide a partial correction to Isbell's claim that every injective metric space is freely contractible, and present a counterexample to a step in the original argument.

Paper Structure

This paper contains 11 sections, 17 theorems, 39 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. Simplical complexes, polyhedra, and collapsibility
  5. Free deformation retraction
  6. Shadows in cylinders
  7. Cylinderwise collapsing
  8. Collapsibility and free deformation retraction
  9. Injective metric and free deformation retraction
  10. Free deformation contractibility of compact injective spaces
  11. Discussion

Key Result

Theorem 1

Let $C$ be a finite cubical complex, and let $d_{p}$ be the metric on $C$ induced by $\ell_{p}$-norms on the cubes of $C$ (considered as unit cubes). Then the following statements are equivalent:

Figures (6)

  • Figure 1: The blue segment $\{p\} \times I$ intersects the principal horizontal simplices $a_i a_{i+1}$ in points and the principal vertical simplices $a_i a_{i+1} a_{i+2}$ along segments.
  • Figure 2: The action of $h_{0.45}$ on a 3-simplex (left) and on a 2-simplex (right). The image of the region containing $B$ is shown in red.
  • Figure 3: The sets $M$ and $N$ inside $P \times I$.
  • Figure 4: Illustration for \ref{['l:path_order']}. The left picture shows $X \times I$ with the sets $Z \subset Y$, $Y \setminus h^{-1}(h(Z))$, and the downward closure of $Y \setminus h^{-1}(h(Z))$. The right picture shows the images of these sets under $h$. The lemma claims that the pink set in the image does not intersect the blue set representing $h(Z)$.
  • Figure 5: A step in the proof of \ref{['theorem:main_theorem']}. The left picture shows the simplex $A_{i}$ inside $|T_{i}|$. The collapse is from the bottom face toward the faces meeting the grey region $|T_{i+1}|$. The right picture shows the images of $A_{i}$ and of the set $L$.
  • ...and 1 more figures

Theorems & Definitions (40)

  • Conjecture : Zeeman conjecture zeeman1963dunce
  • Theorem : melikhov_talk
  • proof
  • Definition
  • Lemma
  • proof
  • Lemma 1
  • proof
  • Definition
  • Lemma 2
  • ...and 30 more