Blow-up groupoid of singular foliations

TL;DR

The paper tackles the challenge that holonomy groupoids for singular foliations are typically non-locally-compact and non-Hausdorff by constructing a blow-up desingularization $\mathcal{H}_{blup}(\mathcal{F})$ with a locally compact, locally Hausdorff structure and Lie algebroid $T\mathcal{F}$. It develops a comprehensive framework combining the blow-up space $\mathrm{blup}(\mathcal{F})$, its tangent bundle $T\mathcal{F}$, the holonomy action, and the blow-up groupoid, establishing smoothness properties and integration results. The work also introduces half-continuous fields of $C^*$-algebras and the $z$-completion, connecting the geometric construction to operator-algebraic decompositions and spectral data via the characteristic set $\mathcal{T}^*\mathcal{F}$. These constructions enable applying noncommutative-geometry techniques to singular foliations, with potential implications for index theory, Dirac operators on singular spaces, and the Baum–Connes program.

Abstract

We introduce a blow-up construction of a smooth manifold along the singular leaves of an arbitrary singular foliation in the sense of Stefan and Sussmann, as well as a blow-up construction of the holonomy groupoid defined by Androulidakis and Skandalis. Our construction gives a locally compact locally Hausdorff groupoid, which can be regarded as a desingularisation of the singular foliation. We show that it retains some smooth structure.

Theorems & Definitions (69)

• Example 1
• Theorem A
• Definition 1.1
• Proposition 1.2
• proof
• Example 1.3
• Theorem 1.4
• Remark 1.5
• Lemma 1.6: Periodic bounding lemma period1period2
• Lemma 1.7: Parametrised periodic bounding lemma