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Rigidity of webs and families of hypersurfaces. (English) Zbl 0583.57015
Singularities and dynamical systems, Proc. Int. Conf., Heraklion/Greece 1983, North-Holland Math. Stud. 103, 271-283 (1985).
[For the entire collection see Zbl 0547.00033.]
A (p$$+1)$$-web of codimension n of $${\mathbb{R}}^{qn}$$ is a $$(p+1)$$-tuple of p to p transversal foliations of codimension n of $${\mathbb{R}}^{qn}$$. The authors demonstrate that the local smooth classification of such webs, with $$p\geq q$$, is the same as the topological one. A one-parameter family of smooth hypersurfaces of $${\mathbb{R}}^ n$$ can be studied by considering the image of a foliated manifold by a smooth map. The authors use the theory developed on webs to prove that local topological stability is not a generic property for one-parameter families of hypersurfaces in $${\mathbb{R}}^ n$$.
Reviewer: J.G.Timourian

##### MSC:
 57R30 Foliations in differential topology; geometric theory 57R45 Singularities of differentiable mappings in differential topology 58C25 Differentiable maps on manifolds 58K99 Theory of singularities and catastrophe theory 57R40 Embeddings in differential topology 32Sxx Complex singularities