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Lifting coarse homotopies

Thomas Weighill

Abstract

Coarse geometry, and in particular coarse homotopy theory, has proven to be a powerful tool for approaching problems in geometric group theory and higher index theory. In this paper, we continue to develop theory in this area by proving a Coarse Lifting Lemma with respect to a certain class of bornologous surjective maps. This class is wide enough to include quotients by coarsely discontinuous group actions, which allows us to obtain results concerning the coarse fundamental group of quotients which are analogous to classical topological results for the fundamental group. As an application, we compute the fundamental group of metric cones over negatively curved compact Riemannian manifolds.

Lifting coarse homotopies

Abstract

Coarse geometry, and in particular coarse homotopy theory, has proven to be a powerful tool for approaching problems in geometric group theory and higher index theory. In this paper, we continue to develop theory in this area by proving a Coarse Lifting Lemma with respect to a certain class of bornologous surjective maps. This class is wide enough to include quotients by coarsely discontinuous group actions, which allows us to obtain results concerning the coarse fundamental group of quotients which are analogous to classical topological results for the fundamental group. As an application, we compute the fundamental group of metric cones over negatively curved compact Riemannian manifolds.

Paper Structure

This paper contains 6 sections, 21 theorems, 34 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Coarse homotopies
  3. Coarse Lifting lemma
  4. Coarse fundamental group
  5. Group actions
  6. Application to metric cones

Key Result

Lemma 2.2

Let $X$ be a geodesic metric space with a decomposition $X = A_1 \cup A_2 \cup \cdots \cup A_N$ where each $A_i$ is a closed set. Let $f: X \to Y$ be a map to a metric space $Y$ such that each restriction $f_i: A_i \to Y$ is coarse with control function $\rho_i$. Then $f$ is coarse.

Figures (1)

  • Figure 1: $c([0,1])$

Theorems & Definitions (57)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Example 2.6
  • Proposition 2.7
  • proof
  • Remark 2.8
  • ...and 47 more