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\(\Gamma\)-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
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