Homotopy invariance of cohomology and signature of a riemannian foliation
arXiv:1703.10448
Abstract
We prove that any smooth foliation that admits a Riemannian foliation structure has a well-defined basic signature, and this geometrically defined invariant is actually a foliated homotopy invariant. We also show that foliated homotopic maps between Riemannian foliations induce isomorphic maps on basic Lichnerowicz cohomology, and that the Alvarez class of a Riemannian foliation is invariant under foliated homotopy equivalence.