[Paper Review] Introduction to the Tangent Grupoid

Alejandro Rivero|ArXiv.org|Oct 23, 1997

Homotopy and Cohomology in Algebraic Topology2 references3 citations

TL;DR

This paper introduces the tangent grupoid as a geometric structure that unifies the tangent bundle and the pair groupoid of a manifold, providing a foundational framework for noncommutative geometry and index theory. It presents plausible definitions and outlines applications in deformation quantization and operator algebras, serving as a pedagogical introduction to the concept.

ABSTRACT

We present some plausible definitions for the tangent grupoid of a manifold M, as well as some of the known applications of the structure. This is a kind of introductory note.

Motivation & Objective

• To define and motivate the tangent grupoid as a geometric object that interpolates between the tangent bundle and the pair groupoid of a manifold.
• To provide a self-contained, introductory exposition suitable for researchers new to the concept.
• To highlight the relevance of the tangent grupoid in noncommutative geometry and deformation theory.
• To establish foundational definitions and structures that enable further applications in index theory and quantization.
• To bridge differential geometry with groupoid theory through a natural geometric construction.

Proposed method

• Construct the tangent grupoid as a topological groupoid over the disjoint union of the manifold and its tangent bundle.
• Define the groupoid structure via a one-parameter family of diffeomorphisms that interpolate between the pair groupoid and the tangent bundle.
• Use the action of the multiplicative group R* to scale tangent vectors and define the groupoid multiplication.
• Equip the tangent grupoid with a topology that makes the source and target maps continuous and the multiplication smooth away from the unit space.
• Demonstrate that the tangent grupoid captures the infinitesimal structure of the manifold through its restriction to the tangent bundle.
• Illustrate the construction via explicit examples on Euclidean space and general manifolds.

Experimental results

Research questions

• RQ1 How can a groupoid structure be defined that naturally interpolates between the pair groupoid and the tangent bundle of a manifold?
• RQ2 What topological and smooth structures are necessary to make the tangent grupoid a well-defined geometric object?
• RQ3 In what way does the tangent grupoid encode the infinitesimal geometry of the underlying manifold?
• RQ4 How does the tangent grupoid facilitate constructions in noncommutative geometry and index theory?
• RQ5 What are the implications of the tangent grupoid for deformation quantization and operator algebras?

Key findings

• The tangent grupoid is constructed as a topological groupoid that unifies the pair groupoid and the tangent bundle via a one-parameter family of diffeomorphisms.
• The groupoid multiplication is well-defined and smooth on the regular part, with the unit space corresponding to the disjoint union of the manifold and its tangent bundle.
• The structure provides a natural framework for studying the symbol map in pseudodifferential operators via the limit at zero scaling.
• The tangent grupoid enables a geometric interpretation of the linearization process in noncommutative geometry.
• The construction is generalizable to arbitrary smooth manifolds and provides a foundation for further developments in index theory.
• The paper establishes the tangent grupoid as a key tool in deformation theory and the study of groupoid C*-algebras.