The twisted higher harmonic signature for foliations

Abstract

We prove that the higher harmonic signature of an even dimensional oriented Riemannian foliation $F$ of a compact Riemannian manifold $M$ 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.