Homotopy characteristic classes of foliations
type: Item
instance of: scholarly article
title: Homotopy characteristic classes of foliations
zbMATH Open document ID: 0687.57010
author: K. Appert
published in: Illinois Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: The secondary classes of foliations are cohomology invariants used to distinguish foliations of manifolds up to concordance. The dual homotopy invariants of foliations are linear functionals on the homotopy groups of a foliated manifold, which again distinguish concordance classes of foliations. This paper addresses the question: Do there exist examples of foliations on a compact manifold which are distinct up to concordance, but have vanishing primary, secondary and dual homotopy invariants? Theorem 7 of the paper under review shows that the answer is yes, with the lowest dimensional examples constructed of codimension 2 on the 9- torus. This is proven by using rational homotopy theory to construct a theory of tertiary characteristic classes for foliations. The tertiary classes are cohomology invariants defined for (restricted sets of) homotopy classes of maps into foliation classifying spaces. Their construction is based on using algebraic Postnikov theory for the minimal model of the secondary algebras WO(q) and W(q). Many other applications are described of these new invariants, including results about the non- triviality of the universal Postnikov invariants for foliation classifying spaces.
Mathematics Subject Classification ID: 57R32
Mathematics Subject Classification ID: 55P62
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 4126326
zbMATH Keywords: Postnikov invariants of classifying spaces of foliations
zbMATH Keywords: secondary classes of foliations
zbMATH Keywords: dual homotopy invariants of foliations
zbMATH Keywords: rational homotopy theory
zbMATH Keywords: tertiary characteristic classes for foliations
zbMATH Keywords: homotopy classes of maps into foliation classifying spaces
zbMATH Keywords: minimal model of the secondary algebras
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4126326