The Twisted Higher Harmonic Signature for Foliations

TL;DR

This paper proves that the higher harmonic signature and twisted higher Betti classes are invariant under leafwise homotopy for certain foliations, with implications for the Novikov conjecture.

Contribution

It establishes leafwise homotopy invariance of the higher harmonic signature and Betti classes in the twisted setting, advancing understanding of foliation invariants.

Findings

Higher harmonic signature is leafwise homotopy invariant.

Twisted higher Betti classes are leafwise homotopy invariant.

Implications for the Novikov conjecture for foliations and groups.

Abstract

We prove that the higher harmonic signature of an even dimensional oriented Riemannian foliation of a compact Riemannian manifold with coefficients in a leafwise U(p,q)-flat complex bundle is a leafwise homotopy invariant. We also prove the leafwise homotopy invariance of the twisted higher Betti classes. Consequences for the Novikov conjecture for foliations and for groups are investigated. Replaces The Higher Harmonic Signature for Foliations I: The Untwisted Case, and contains significant improvements.