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The Atiyah–Patodi–Singer signature formula for measured foliations

Published/Copyright: March 7, 2013
Journal für die reine und angewandte Mathematik (Crelles Journal)
From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2014 Issue 695

Abstract

Given a compact manifold with boundary X0 endowed with a foliation ℱ0 transverse to the boundary, and which admits a holonomy invariant transverse measure Λ, we define three types of signature for the pair (foliation, boundary foliation): the analytic signature, denoted by σΛ,an(X0,∂X0), is the leafwise L2-Λ-index of the signature operator on the extended manifold X obtained by attaching cylindrical ends to the boundary; the Hodge signature σΛ,Hodge(X0,∂X0) is defined using the natural representation of the Borel groupoid ℛ of X on the field of square integrable harmonic forms on the leaves; and the de Rham signature, σΛ,dR(X0,∂X0), defined using the natural representation of the Borel groupoid ℛ0 of X0 on the field of the L2-relative de Rham spaces of the leaves. We prove that these three signatures coincide

σΛ,an(X0,∂X0)=σΛ,Hodge(X0,∂X0)=σΛ,dR(X0,∂X0).

As a consequence of the index formula we proved in [Bull. Sci. Math. (2010), DOI 10.1016/ j.bulsci.2010.10.003], we finally obtain the Atiyah–Patodi–Singer signature formula for measured foliations:

σΛ,dR(X0,∂X0)=〈L(Tℱ0),CΛ〉+1/2[ηΛ(Dℱ∂)].

The author wishes to thank Paolo Piazza for having suggested the problem and for a number of interesting discussions, Georges Skandalis, James Heitsch, Moulay T. Benameur, Sara Azzali, Eric Leichtnam, Stéphane Vassout and Yuri Kordyukov for discussions and comments. We heartily thank the referee for pointing out a wrong proof and suggesting the solution.

Received: 2012-11-20
Revised: 2012-12-10
Published Online: 2013-3-7
Published in Print: 2014-10-1

© 2014 by De Gruyter

Articles in the same Issue

  1. Frontmatter
  2. Comparison of dualizing complexes
  3. Variational principles for immersed surfaces with L2-bounded second fundamental form
  4. A comparison of Paley–Wiener theorems for real reductive Lie groups
  5. Seshadri constants via toric degenerations
  6. Central sequence C*-algebras and tensorial absorption of the Jiang–Su algebra
  7. Erratum to Central sequence C*-algebras and tensorial absorption of the Jiang–Su algebra (J. reine angew. Math. 695 (2014), 175–214)
  8. The Atiyah–Patodi–Singer signature formula for measured foliations
https://doi.org/10.1515/crelle-2012-0117

Articles in the same Issue

  1. Frontmatter
  2. Comparison of dualizing complexes
  3. Variational principles for immersed surfaces with L2-bounded second fundamental form
  4. A comparison of Paley–Wiener theorems for real reductive Lie groups
  5. Seshadri constants via toric degenerations
  6. Central sequence C*-algebras and tensorial absorption of the Jiang–Su algebra
  7. Erratum to Central sequence C*-algebras and tensorial absorption of the Jiang–Su algebra (J. reine angew. Math. 695 (2014), 175–214)
  8. The Atiyah–Patodi–Singer signature formula for measured foliations
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