Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$H_0(C_c(\mathcal{G}_\bullet,\mathbb{Z}))\neq H_0^{\mathrm{sing}}(B\mathcal{G};\mathbb{Z})$ for the Cantor unit groupoid

Luciano Melodia

Abstract

For an ample groupoid $\mathcal{G}$, Matui type groupoid homology is computed from the nerve $\mathcal{G}_\bullet$ via Moore chains $C_c(\mathcal{G}_n,\mathbb{Z})$ and the alternating sum of pushforwards along the face maps. We give an explicit example showing that this homology need not agree with singular homology of the classifying space $B\mathcal{G}$. The discrepancy occurs already in degree $0$ for the unit groupoid on the Cantor set.

$H_0(C_c(\mathcal{G}_\bullet,\mathbb{Z}))\neq H_0^{\mathrm{sing}}(B\mathcal{G};\mathbb{Z})$ for the Cantor unit groupoid

Abstract

For an ample groupoid , Matui type groupoid homology is computed from the nerve via Moore chains and the alternating sum of pushforwards along the face maps. We give an explicit example showing that this homology need not agree with singular homology of the classifying space . The discrepancy occurs already in degree for the unit groupoid on the Cantor set.
Paper Structure (5 sections, 9 theorems, 28 equations)

This paper contains 5 sections, 9 theorems, 28 equations.

Table of Contents

  1. Introduction and statement
  2. Pushforward and Moore chains
  3. The Cantor unit groupoid and Moore homology
  4. Classifying space and singular $H_0$
  5. Cardinality separation

Key Result

Proposition 1.1

Let $X$ be the Cantor set and let $\mathcal{G}\coloneqq (X\rightrightarrows X)$ be the unit groupoid. Then whereas In particular $H_0(\mathcal{G};\mathbb Z)\not\cong H_0^{\mathrm{sing}}(B\mathcal{G};\mathbb Z)$.

Theorems & Definitions (19)

  • Proposition 1.1
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 4.1
  • proof
  • ...and 9 more