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Transverse transversals and homeomorphic transversals. (English) Zbl 0574.57013
Let X, Y be disjoint $$C^ 1$$ submanifolds of $${\mathbb{R}}^ n$$ and let $$0\in Y\cap \bar X$$. The pair (X,Y) is said to have transverse $$C^ k$$ transversals of dimension s at 0 (1$$\leq k\leq \infty$$, cod $$Y\leq s\leq n- 1)$$ if $$(t^ k_ s):$$ for every $$C^ k$$ submanifold S of dimension s transverse to Y at 0 there is some neighbourhood of 0 in which S is transverse to X. We say that (X,Y) has homeomorphic $$C^ k$$ transversals of dimension s at 0 if $$(h^ k_ s):$$ given $$C^ k$$ submanifolds $$S_ 1,S_ 2$$ of dimension s transverse to Y at 0, the germs at 0 of $$S_ 1\cap X$$ and $$S_ 2\cap X$$ are homeomorphic.
In this paper the author studies relations of these conditions. He shows that $$(h^ k_ s)$$ implies $$(t^ k_ s)$$ and $$(t^ 1_ s)$$ implies $$(h^ 1_ s)$$. But examples show that $$(t^ k_ s)$$ does not imply $$(h^ k_ s)$$ if $$k\geq 2$$ nor does $$(t^ k_ s)$$ imply $$(t_ s^{k- 1})$$, nor $$(h^ k_ s)$$ imply $$(h_ s^{k-1})$$. He also shows that $$(t^ 1_ s)$$ and $$(t^ 1_{s+1})$$ are equivalent except when $$s=cod Y$$ and makes precise when $$(t^ 1_ s)$$ and Whitney (a) are equivalent. These results are closely related to characterisation of V-sufficiency of jets.
Reviewer: S.Izumiya

##### MSC:
 57R40 Embeddings in differential topology 57N80 Stratifications in topological manifolds 58C25 Differentiable maps on manifolds 58K99 Theory of singularities and catastrophe theory 57R45 Singularities of differentiable mappings in differential topology 58A35 Stratified sets
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