The supplied smooth Godbillon-Vey construction does not cover merely C1 foliations

Assume Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$. The construction in The Godbillon-Vey class of a codimension-one foliation is stated for smooth foliations and smooth defining data. It supplies no cohomological Godbillon–Vey class for merely $C^1$ foliations.

This is a limitation of the supplied construction, not a necessary regularity threshold for every Godbillon–Vey theory. Hurder–Katok, §7, Proposition 7.1, constructs a natural Godbillon–Vey invariant for transversally $C^{1,\alpha }$ codimension-one foliations of closed oriented $3$-manifolds when $\alpha >1\mathrm{/}2$, extending the $C^2$ invariant. Their argument uses a distributional pairing; it is not the smooth differential-form construction supplied here.

The usual finite-regularity theory defines the Godbillon–Vey class for $C^2$ foliations and the Godbillon measure for $C^1$ foliations. Hurder–Langevin, §3, printed p.10, states this distinction, then explicitly specializes §3.1 to smooth foliations and refers elsewhere for the required finite-regularity modifications. Those modifications are not proved on this page. In particular one cannot obtain the $C^2$-atlas theory merely by replacing “smooth” by “$C^2$” in the smooth-form proof: the associated defining form may have lower regularity, and comparison with smooth de Rham cohomology requires additional work.