##
*
V. V. Shurygin
*

The structure of smooth mappings over Weil algebras and the
category of manifolds over algebras

*(Lobachevskii Journal of Mathematics,
Vol.5, pp.29-55)*

As is known, the bundle $T^\A M_n$ of infinitely near points of
$\A$-type defined for any local Weil algebra $\A$ and smooth real
manifold $M_n$ is one of basic examples of smooth manifolds over $\A$.
In the present paper we give a description of the local structure of
smooth mappings in the category of smooth manifolds over local
algebras and consider various examples of such manifolds.
Next we study the homotopy and holonomy groupoids of a
smooth manifold $M^\A_n$ over a local algebra $\A$ associated with canonical
foliations corresponding to ideals of $\A$.
In particular, it is proved that a complete manifold $M^\A_n$ has
neither homotopy nor holonomy vanishing cycles.

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