On (\(\omega\) ) and \((t^ s)\)-regular stratifications
Let \(G_{n,d}\) denote the Grassmannian of d-planes in \(R^ n\), and let \[ \pi_{n,d}: E_{n,d}\to G_{n,d} \] be the canonical d-plane bundle. Here, \[ E_{n,d}=\{(L,x)| \quad x\in L\in G_{n,d}\}. \] We call the projection \(\beta_{n,d}: E_{n,d}\to R^ n\) a \(G_{n,d}\)-blowing up of \(R^ n\). Let X, Z be two disjoint \(C^ s\)- submanifolds of \(R^ n\) in a neighbourhood of 0, \(0\in Z\), such that dim Z\(=c\) and \(d=n-c.\) We say that X is \(t^ s\)-regular over Z at 0 if every \(C^ s\)-submanifold, of dimension d, which is transverse to X at 0, is transverse to X near 0. We define \[ \tilde X=\{(L,x)| \quad x\in X,\quad L,\quad Z\quad transverse\quad at\quad 0\},\quad \tilde Z=\{(L,0)| \quad L,\quad Z\quad transverse\quad at\quad 0\}. \] The authors state the following main result. Theorem 1. Suppose Z is analytic and X is subanalytic; \(s\geq 0\). Then X is \(t^{s+1}\)-regular over Z at 0 if and only if \(\tilde X\) is \(t^ s\)- regular over \(\tilde Z.\) As a consequence of this theorem and the Thom-Mather-Verdier Isotopy Theorem, the conjecture of Thom on transversals to an (a)-regular pair is proved for the subanalytic case. On the other hand, the authors have established an analytic criterion for \(t^ s\)-regularity (Theorem 2), an analogue of the Kuo's Ratio Test. This criterion is used to prove Theorem 1.
scientific article; zbMATH DE number 4065942