Foliation surgeries and the concordance groups of foliated spheres
Benjamin B. McMillan
TL;DR
The paper develops a unified framework to extend manifold surgery to foliated spaces via Haefliger structures, introducing weak, lax, and strong foliation surgeries and an obstruction theory controlled by the restricted Haefliger structure on attaching spheres. It proves that for attaching spheres of dimension up to $q+1$ with trivial normal bundle, obstructions vanish and strong surgeries exist, enabling sequences of foliation surgeries that reduce to a Haefliger structure on the sphere $S^{n}$ with unchanged characteristic numbers when $M$ is stably trivial and $n \le 2q+2$. This yields a group structure on concordance classes of foliated spheres, identified with $\pi_{k}(B\Gamma^{+}_{q})$, and connects these classes to the Haefliger classifying spaces and their characteristic classes, proving surjectivity results for $\pi_{2q+1}(B\Gamma^{+}_{q})$ onto spaces of positively dimensional Gel'fand–Fuks invariants. The approach also provides explicit constructions of foliations on $(2q+1)$-spheres whose Godbillon–Vey numbers realize continuous variation, and adapts Thurston’s construction to yield explicit GV-spheres; a mechanism to unclasp the normal bundle demonstrates how obstructions can be removed in significant cases. Overall, the work strengthens the bridge between foliation theory, Haefliger structures, and high-dimensional surgery, with concrete consequences for the spherical realization of independent characteristic classes and for constructing explicit foliations with prescribed invariants on spheres.
Abstract
We define a general procedure extending surgery to manifolds with foliation or Haefliger structure, the resulting Haefliger structure defined on the underlying surgered manifold. We find a single obstruction, whose vanishing characterizes the existence of a foliation surgery along an attaching sphere. When unobstructed, the resulting Haefliger structure can be chosen so that characteristic numbers are unchanged. Moreover, the obstructions are highly tractable for attaching spheres of dimension small relative the foliation codimension. This allows us to show that on every stably trivial manifold of dimension \( n \le 2q+2 \), a codimension-\( q \) Haefliger structure with trivial normal bundle admits a sequence of foliation surgeries to a Haefliger structure on the sphere \( S^{n} \), characteristic numbers unchanged. The obstruction always vanishes for \( 0 \)-sphere surgeries. It follows that the foliation connected sum gives, for each sphere \( S^{n} \) and codimension \( q \), a group structure on the set of concordance classes of transversely oriented Haefliger structures on \( S^{n} \). The resulting groups may be identified with the homotopy groups of the Haefliger classifying spaces \( BΓ^{+}_{q} \). Using this, we recover a result of Hurder that \( π_{2q+1}(BΓ^{+}_{q}) \) surjects onto a vector space of positive dimension for each codimension \( q \). We also apply the techniques here to a construction of Thurston's, to construct explicit Haefliger structures on \( (2q+1) \)-spheres whose Godbillon-Vey numbers surject to the real line.