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Radiance obstructions for smooth manifolds over Weil algebras. (English, Russian) Zbl 1120.58002
Russ. Math. 49, No. 5, 67-79 (2005); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2005, No. 5, 71-83 (2005).
A finite-dimensional commutative associative \(\mathbb R\)-algebra \(\mathbb A\) with unity is a Weil algebra if its radical \(\overset {_{\circ}}{\mathbb A}\) is the only maximal ideal and the quotient algebra \(\mathbb A/\overset {_{\circ}}{\mathbb A}\) is isomorphic to \(\mathbb R\).
In this paper, the author defines a manifold \(M^{\mathbb A}_ n\) formed from the atlas of \(M\) by means of \(\mathbb A\)-diffeomorphisms between open subsets of modules \(\mathbb A^ n\) and constructs cohomology classes with coefficients in some bundles associated to \(M^{\mathbb A}_ n\) which form obstructions for the existence of a bundle structure on \(M^{\mathbb A}_ n\to M\). If \(M^{\mathbb A}_ n\) is complete, the constructed classes are trivial if and only if \(M^{\mathbb A}_ n\) is isomorphic to the Weil bundle \(T^{\mathbb A}M\).

58A32 Natural bundles
13M05 Structure of finite commutative rings