Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A transverse index theorem in the calculus of filtered manifolds

Clément Cren

TL;DR

The paper develops a transverse index theory for filtered manifolds by introducing transverse symbols relative to an integrable subbundle and a transversally Rockland condition. It builds a KK-theory framework using both symbol-based and pseudodifferential constructions, and proves a Poincaré-type duality linking the K-homology class of an operator to the transverse symbol. Central tools include deformation groupoids, holonomy actions, and equivariant KK-theory, which together yield a transverse index that generalizes elliptic indices to hypoelliptic, filtered settings. The results encompass descent to equivariant indices and an explicit link between symbol and operator cycles, with potential applications to fibred foliations and group actions on filtered spaces.

Abstract

We use filtrations of the tangent bundle of a manifold starting with an integrable subbundle to define transverse symbols to the corresponding foliation, define a condition of transversally Rockland and prove that transversally Rockland operators yield a K-homology class. We construct an equivariant KK-class for transversally Rockland transverse symbols and show a Poincare duality type result linking the class of an operator and its symbol.

A transverse index theorem in the calculus of filtered manifolds

TL;DR

The paper develops a transverse index theory for filtered manifolds by introducing transverse symbols relative to an integrable subbundle and a transversally Rockland condition. It builds a KK-theory framework using both symbol-based and pseudodifferential constructions, and proves a Poincaré-type duality linking the K-homology class of an operator to the transverse symbol. Central tools include deformation groupoids, holonomy actions, and equivariant KK-theory, which together yield a transverse index that generalizes elliptic indices to hypoelliptic, filtered settings. The results encompass descent to equivariant indices and an explicit link between symbol and operator cycles, with potential applications to fibred foliations and group actions on filtered spaces.

Abstract

We use filtrations of the tangent bundle of a manifold starting with an integrable subbundle to define transverse symbols to the corresponding foliation, define a condition of transversally Rockland and prove that transversally Rockland operators yield a K-homology class. We construct an equivariant KK-class for transversally Rockland transverse symbols and show a Poincare duality type result linking the class of an operator and its symbol.
Paper Structure (18 sections, 26 theorems, 74 equations)

This paper contains 18 sections, 26 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Geometric preliminaries
  3. The setting and the holonomy action
  4. Deformation groupoids
  5. Symbolic calculus
  6. Symbols in the filtered calculus
  7. Restriction of the symbol
  8. Equivariance of distributions
  9. Transverse cycle for transversally Rockland operators
  10. The KK-cycle of the symbol
  11. The equivariant KK-cycle
  12. Descent and index
  13. Symbols of positive order
  14. The pseudodifferential construction
  15. The van Erp Yunken picture of pseudodifferential calculus
  16. ...and 3 more sections

Key Result

Lemma 2.1

Let $G \rightrightarrows M$ be a Lie groupoid, and $Y \subset M$ a submanifold. Let $\rho$ be the anchor map of the associated algebroid $\mathcal{A} G \to M$. If for all $y \in Y,$$\rho(\mathcal{A}_yG) \subset T_yY$ then $G\cdot Y \subset Y,$ i.e. $\forall \gamma \in G, s(\gamma) \in Y \Leftrightar

Theorems & Definitions (67)

  • Lemma 2.1
  • proof
  • Corollary 2.1.1
  • proof
  • Corollary 2.1.2
  • proof
  • Example 2.2
  • Remark 2.3
  • Example 2.4
  • Example 2.5
  • ...and 57 more