The A.P.S. signature formula for measured foliations

TL;DR

This paper introduces and compares different types of signatures for foliated manifolds with boundary, establishing their equivalence and deriving a Hirzebruch formula to connect them.

Contribution

It defines the Analytical, Hodge, and de Rham signatures for foliated manifolds with boundary and proves their equivalence along with a Hirzebruch formula.

Findings

All signatures coincide for the considered foliated manifolds.

A Hirzebruch formula relating these signatures is established.

The signatures are well-defined for foliated manifolds with boundary.

Abstract

We define the Analytical signature, the Hodge signature and the de Rham signature for a foliated manifold with boundary with foliation transverse to the boundary. We show that all these signatures coincide and a Hirzebruch formula is valid.