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Homotopy invariance of cohomology and signature of a riemannian foliation

Georges Habib, Ken Richardson

Abstract

We prove that any smooth foliation that admits a Riemannian foliation structure has a well-defined basic signature, and this geometrically defined invariant is actually a foliated homotopy invariant. We also show that foliated homotopic maps between Riemannian foliations induce isomorphic maps on basic Lichnerowicz cohomology, and that the Alvarez class of a Riemannian foliation is invariant under foliated homotopy equivalence.

Homotopy invariance of cohomology and signature of a riemannian foliation

Abstract

We prove that any smooth foliation that admits a Riemannian foliation structure has a well-defined basic signature, and this geometrically defined invariant is actually a foliated homotopy invariant. We also show that foliated homotopic maps between Riemannian foliations induce isomorphic maps on basic Lichnerowicz cohomology, and that the Alvarez class of a Riemannian foliation is invariant under foliated homotopy equivalence.

Paper Structure

This paper contains 10 sections, 25 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Riemannian foliations
  4. Twisted basic cohomology
  5. The basic signature operator
  6. Known results on the homotopy invariance of the basic cohomology and signature in the taut case
  7. Homotopy invariance of twisted cohomology and basic signature
  8. Twisted basic cohomology and basic signature for general Riemannian foliations
  9. Basic Lichnerowicz cohomology and foliated homotopy invariance
  10. Invariance of the basic signature

Key Result

Theorem \oldthetheorem

(In HabRi) Let $(M,\mathcal{F})$ be a compact Riemannian manifold endowed with a Riemannian foliation and basic Clifford bundle $E\rightarrow M$. The spectrum of the basic Dirac operator is the same for every possible choice of bundle-like metric that is associated to the transverse metric on the qu

Theorems & Definitions (46)

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  • ...and 36 more