Leafwise homotopies and Hilbert-Poincaré complexes I. Regular HP-complexes and leafwise pull-back maps

Abstract

In the first part of our series of papers, we prove the leafwise homotopy invariance of K-theoretic signatures of foliations and laminations, using the formalism of Hilbert–Poincaré complexes as revisited by Higson and Roe. We use a generalization of their methods to give a homotopy equivalence of de Rham–Hilbert–Poincaré complexes associated with leafwise homotopy equivalence of foliations. In particular, we obtain an explicit path connecting the signature classes in K-theory, up to isomorphism induced by Morita equivalence. Applications of this path on the stability properties of rho-invariants à la Keswani will be carried out in the later parts of this series.