Foliation surgeries and the concordance groups of foliated spheres

TL;DR

This paper develops a surgery procedure for manifolds with foliations, identifies obstructions, and constructs groups of foliation structures on spheres, linking them to homotopy groups of classifying spaces.

Contribution

It introduces a general foliation surgery method, characterizes obstructions, and establishes a group structure on foliation classes related to Haefliger spaces.

Findings

A single obstruction to foliation surgery along an attaching sphere.

Surgered structures preserve characteristic numbers when unobstructed.

Groups of foliation structures on spheres are isomorphic to homotopy groups of classifying spaces.

Abstract

We define a general procedure extending surgery to manifolds with foliation or Haefliger structure. We find a single obstruction to foliation surgery along an attaching sphere. When unobstructed, the surgery can be chosen to preserve characteristic numbers. Studying these obstructions, we obtain two results. First, on every stably trivial manifold of dimension \( n \le 2q+2 \), a transversely framed codimension-\( q \) Haefliger structure can be surgered to a Haefliger structure on the sphere \( S^{n} \), characteristic numbers unchanged. As an application, we modify a construction of Thurston's to give explicit Haefliger structures on \( S^{2q+1} \) whose Godbillon-Vey numbers surject to the real line. Second, the foliation connected sum gives, for each dimension \( n \) and codimension \( q \), a group structure on concordance classes of transversely oriented Haefliger structures on \( S^{n} \). These groups may be identified with the homotopy groups of the Haefliger classifying spaces \( B\Gamma^{+}_{q} \).