Transverse transversals and homeomorphic transversals (Q1063912)

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.