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Rigid secondary characteristic classes

Steven Hurder

Abstract

We construct families of non-trivial universal rigid secondary classes for foliations, and then discuss their application to prove that foliations are not homotopic. An observation of Lawson about the non-triviality of the normal Pontrjagin classes of foliations is extended, and then used to construct new families of examples of foliations with non-trivial rigid secondary classes. Examples are given of (abstractly constructed) foliations of compact manifolds with homotopic tangent bundles, but which are not homotopic as foliations.

Rigid secondary characteristic classes

Abstract

We construct families of non-trivial universal rigid secondary classes for foliations, and then discuss their application to prove that foliations are not homotopic. An observation of Lawson about the non-triviality of the normal Pontrjagin classes of foliations is extended, and then used to construct new families of examples of foliations with non-trivial rigid secondary classes. Examples are given of (abstractly constructed) foliations of compact manifolds with homotopic tangent bundles, but which are not homotopic as foliations.
Paper Structure (5 sections, 13 theorems, 40 equations)

This paper contains 5 sections, 13 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Pontrjagin classes
  3. Rigid Secondary classes
  4. Spherical classes
  5. Applications

Key Result

THEOREM 1.3

The classes in ${\mathcal{R}}_q$ are invariants of the smooth homotopy class of a foliation; those in $\overline{\mathcal{R}}_q$ are invariants of the smooth homotopy class of a foliation with framed normal bundle.

Theorems & Definitions (15)

  • THEOREM 1.3
  • THEOREM 1.4
  • DEFINITION 1.5
  • THEOREM 1.6
  • THEOREM 2.1
  • THEOREM 2.2
  • THEOREM 3.1: Strong Bott Vanishing BottHaefliger1972
  • THEOREM 3.2
  • PROPOSITION 3.3
  • DEFINITION 3.4
  • ...and 5 more