Surgery On Foliations

TL;DR

This paper develops two new surgery theories and Whitehead torsion concepts for foliations, linking topological classification of leaves with operator algebra index theory, and proposes Borel conjectures for foliations.

Contribution

It introduces bounded and bounded geometry surgery theories for foliations, connecting topology, geometry, and operator algebras, and addresses Borel conjectures in this context.

Findings

Constructed bounded surgery theory and Whitehead torsion for foliations.

Established analogy between surgery theory and index theory in operator algebras.

Verified Borel conjectures for certain geometrical cases.

Abstract

In this paper, we set up two surgery theories and two kinds of Whitehead torsion for foliations. First, we construct a bounded surgery theory and bounded Whitehead torsion for foliations, which correspond to the Connes' foliation algebra in the K-theory of operator algebras, in the sense that there is an analogy between surgery theory and index theory, and a Novikov Conjecture for bounded surgery on foliations in analogy with the foliated Novikov conjecture of P.Baum and A.Connes in operator theory. This surgery theory classifies the leaves topologically. Secondly, we construct a bounded geometry surgery for foliations, which is a generalization of blocked surgery, and a bounded geometry Whitehead torsion. The classifications in this surgery theory include the specification of the Riemannian metrics of the leaves up to quasi=isometry. We state Borel conjectures for foliations, which solves a problem posed by S.Weinberger \cite{Wein}, and verify these in some cases of geometrical interest.