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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 $$T^{\mathbb A}M$$.

##### MSC:
 58A32 Natural bundles 13M05 Structure of finite commutative rings