## $$\Gamma$$-foliations and Weil prolongations.(English)Zbl 0815.57021

Bureš, J. (ed.) et al., The proceedings of the winter school geometry and topology, Srní, Czechoslovakia, January 1992. Palermo: Circolo Matemático di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 32, 69-79 (1994).
Let $${\mathcal A}$$ be a local algebra over $$\mathbb{R}$$, so that $${\mathcal A} = \mathbb{R} [X_ 1, X_ 2, \dots, X_ p]/a$$ where $$a$$ is an ideal of finite codimension. Two $$C^ \infty$$ maps $$\varphi, \psi : \mathbb{R}^ p \to M$$ are $${\mathcal A}$$-equivalent if $$\xi_{\mathcal A} \tau (f \varphi) = \xi_{\mathcal A} \tau (f \psi)$$ for each $$f : M \to \mathbb{R}$$, where $$\tau$$ picks out the formal Taylor series and $$\xi_{\mathcal A} : \mathbb{R} [X_ 1, X_ 2, \dots, X_ p] \to {\mathcal A}$$ is the quotient map. Then the set of all $${\mathcal A}$$-equivalence classes of maps $$\mathbb{R}^ p \to M$$ is the $${\mathcal A}$$-prolongation of $$M$$, denoted $$M^{\mathcal A}$$. Given a foliation $${\mathcal F}$$ on a manifold $$M$$ it is shown how to construct the $${\mathcal A}$$- prolongation of $${\mathcal F}$$ on $$M^{\mathcal A}$$. It is shown that if the $${\mathcal A}$$-prolongations of two foliations are homotopic then their sets of characteristic classes are the same.
For the entire collection see [Zbl 0794.00022].

### MSC:

 57R30 Foliations in differential topology; geometric theory 58A20 Jets in global analysis