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A coarse invariant for all metric spaces

Michael DeLyser, Brendon LaBuz, Benjamin Wetsell

Abstract

"An invariant of metric spaces under bornologous equivalences" gives an invariant and "A coarse invariant" extends the invariant to coarse equivalences. In both papers the invariant is defined for a class of metric spaces called sigma stable. This paper extends the invariant to all metric spaces and also gives an example of a space that is not sigma stable.

A coarse invariant for all metric spaces

Abstract

"An invariant of metric spaces under bornologous equivalences" gives an invariant and "A coarse invariant" extends the invariant to coarse equivalences. In both papers the invariant is defined for a class of metric spaces called sigma stable. This paper extends the invariant to all metric spaces and also gives an example of a space that is not sigma stable.

Paper Structure

This paper contains 4 sections, 8 theorems, 1 figure.

Table of Contents

  1. Introduction
  2. The invariant
  3. A space that is not $\sigma$-stable
  4. Direct limits

Key Result

Theorem 1.1

Suppose $f:X\to Y$ is a bornologous equivalence between metric spaces. Let $x_0$ be a basepoint of $X$ and set $y_0=f(x_0)$. Suppose $X$ and $Y$ are $\sigma$-stable. Then $\sigma(X,x_0)=\sigma(Y,y_0)$.

Figures (1)

  • Figure 1: The two sequences are equivalent.

Theorems & Definitions (16)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 6 more