Chern-Weil theory for Haefliger-singular foliations
Lachlan Ewen MacDonald, Benjamin McMillan
TL;DR
This work extends Bott's Chern-Weil framework to Haefliger-singular foliations by introducing adapted geometries on the regular locus and extending the resulting Chern-Weil forms to the whole manifold. Central to the approach is a smooth Haefliger classifying picture via a diffeological space $B\Gamma_q$ and Haefliger bundles, which yields a universal characteristic map from Gel'fand-Fuks cohomology $H^{*}(A(\frak a_q),O(q))$ to de Rham cohomology. The authors prove functoriality and homotopy-invariance of the construction, show that every Haefliger structure is homotopic to a Haefliger-singular foliation with adapted geometry, and extend the Godbillon-Vey algorithm to the singular setting. Together, these results enable differential-geometric analyses of Haefliger structures, with potential applications in noncommutative geometry and index theory, by connecting universal, classifying-space perspectives with explicit geometric representatives of characteristic classes.
Abstract
We give a Chern-Weil map for the Gel'fand-Fuks characteristic classes of Haefliger-singular foliations, those foliations defined by smooth Haefliger structures with dense regular set. Our characteristic map constructs, out of singular geometric structures adapted to singularities, explicit forms representing characteristic classes in de Rham cohomology. The forms are functorial under foliation morphisms. We prove that the theory applies, up to homotopy, to general smooth Haefliger structures: subject only to obvious necessary dimension constraints, every smooth Haefliger structure is homotopic to a Haefliger-singular foliation, and any morphism of Haefliger structures is homotopic to a morphism of Haefliger-singular foliations. As an application, we provide a generalisation to the singular setting of the classical construction of forms representing the Godbillon-Vey invariant.