Characteristic classes for Riemannian foliations

TL;DR

This paper surveys and presents new results on the non-triviality of characteristic classes in Riemannian foliations, including independence of Pontrjagin classes and a negative solution to a conjecture about classifying space maps.

Contribution

It provides new examples of independent Pontrjagin classes and disproves a conjecture regarding the triviality of a classifying space map for codimension greater than one.

Findings

Primary Pontrjagin classes are linearly independent in certain examples.

Secondary classes' independence and total variation are analyzed.

A negative solution to a conjecture about classifying space maps is provided.

Abstract

The purpose of this paper is to both survey and offer some new results on the non-triviality of the characteristic classes of Riemannian foliations. We give examples where the primary Pontrjagin classes are all linearly independent. The independence of the secondary classes is also discussed, along with their total variation. Finally, we give a negative solution of a conjecture that the map of classifying spaces $\FRGq \to \FGq$ is trivial for codimension $q > 1$.