Local topological models of envelopes (Q1093964)

Let \(X^ n\) and \(Y^ n\) be smooth manifolds and \(\Gamma\subset X\times Y\) a smooth hypersurface such that the projections \(\pi_ X: \Gamma \to X\) and \(\pi_ Y: \Gamma \to Y\) are submersions. For any submanifold \(M\subset X\), let \(E(M)=\{y\in Y|\exists x\in M\) so that (x,y)\(\in \Gamma\) and \(T_ xM\times \{0\}\subset T_{(x,y)}\Gamma \}\). Then E(M) is the envelope of the submanifolds \(\pi_ Y\pi_ X^{-1}(x)\) of Y (x\(\in M)\). The principal result of this paper provides local models of E(M), viz for a residual set of embeddings \(M\to X\), local models of E(M) are given by critical values of MT-stable map germs. (A map germ is MT- stable if it is transverse to the canonical stratification of the jet space.)