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A twist over a minimal étale groupoid that is topologically nontrivial over the interior of the isotropy

Becky Armstrong, Abraham C. S. Ng, Aidan Sims, Yumiao Zhou

TL;DR

The paper resolves whether a twist over a minimal etale groupoid must arise from a continuous $2$-cocycle by constructing a twist $\mathcal{E}$ over a minimal groupoid $\mathcal{G}$ built from a nontrivial principal $T$-bundle and a minimal action of the free group $F_2$. It provides a detailed assembly of $\mathcal{E}$ via weight-indexed bundles $B^{(w)}$ over $X$, balanced fibre products, and a coherent inversion, showing $\mathcal{E}$ is a twist over $\mathcal{G}$. The key result is that the induced twist over the interior of isotropy cannot be realized by a continuous $2$-cocycle, as this would imply a global section of a nontrivial bundle, a contradiction. This yields a concrete obstruction in the minimal étale setting and informs the behavior of twisted groupoid C*-algebras and related injectivity theorems.

Abstract

We present an example of a twist over a minimal Hausdorff étale groupoid such that the restriction of the twist to the interior of the isotropy is not topologically trivial; that is, the restricted twist is not induced by a continuous 2-cocycle.

A twist over a minimal étale groupoid that is topologically nontrivial over the interior of the isotropy

TL;DR

The paper resolves whether a twist over a minimal etale groupoid must arise from a continuous -cocycle by constructing a twist over a minimal groupoid built from a nontrivial principal -bundle and a minimal action of the free group . It provides a detailed assembly of via weight-indexed bundles over , balanced fibre products, and a coherent inversion, showing is a twist over . The key result is that the induced twist over the interior of isotropy cannot be realized by a continuous -cocycle, as this would imply a global section of a nontrivial bundle, a contradiction. This yields a concrete obstruction in the minimal étale setting and informs the behavior of twisted groupoid C*-algebras and related injectivity theorems.

Abstract

We present an example of a twist over a minimal Hausdorff étale groupoid such that the restriction of the twist to the interior of the isotropy is not topologically trivial; that is, the restricted twist is not induced by a continuous 2-cocycle.
Paper Structure (9 sections, 11 theorems, 92 equations)

This paper contains 9 sections, 11 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Kumjian's example
  4. The proof of Theorem A
  5. Defining the twisted groupoid as a topological space
  6. Defining multiplication on the twisted groupoid
  7. Defining inversion on the twisted groupoid
  8. Defining the quotient groupoid G and proving that E is a twist over G
  9. Studying the isotropy and proving the desired properties

Key Result

Theorem A

There exists a twist $\mathcal{E}$ over a minimal Hausdorff étale groupoid $\mathcal{G}$ such that the induced twist $\mathcal{I}^\mathcal{E}$ over the interior $\mathcal{I}^\mathcal{G}$ of the isotropy of $\mathcal{G}$ does not come from a $2$-cocycle.

Theorems & Definitions (25)

  • Theorem A
  • Definition 2.1
  • Remark 2.3
  • Remark 2.4
  • Proposition 4.2
  • Lemma 4.3
  • proof
  • proof : Proof of \ref{['proposition: Bw principal T-bundle']}
  • Lemma 4.4
  • proof
  • ...and 15 more