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

TL;DR

This paper extends the theory of eta invariants for laminations by developing regular Hilbert-Poincare complexes and leafwise pull-back maps, incorporating Morita equivalence, and establishing functoriality and homotopy invariance results.

Contribution

It introduces a framework for regular HP-complexes and leafwise pull-back maps that account for Morita equivalence, advancing the study of eta invariants in lamination theory.

Findings

Constructed leafwise pull-back morphisms for de Rham HP complexes.

Proved functoriality of these pull-back maps under leafwise maps.

Established a path connecting leafwise signature operators in homotopy equivalence cases.

Abstract

This paper is a first of a series of three papers which study eta invariants for laminations. In this first paper, we extend the results of Higson and Roe to deal with regular (unbounded) operators and more importantly to take into account Morita equivalence of underlying $C^*$-algebras. Given an oriented leafwise map between leafwise oriented laminations on compact spaces, satisfying some natural assumptions, we construct a pull-back morphism between the de Rham HP complexes and prove their expected functoriality. As a byproduct, when two leafwise oriented laminations are leafwise homotopy equivalent, our construction allows to deduce an explicit path joining the leafwise signature operators and hence the famous Large Time Path appearing in the proof of the homotopy invariance of the measured Cheeger-Gromov number.