Generic bifurcations of varieties. II (Q1099853)

Let f: (R\({}^ n\times R^ r,0)\to (R^ p,0)\) be a smooth map germ. For each u in \(R^ r\) near 0, consider the variety \(f^{-1}(0)\cap (R^ n\times u)\). The author studies a topological classification theory of bifurcations of these varieties. In a previous paper [Part I, ibid. 46, 137-164 (1984; Zbl 0537.58009)] a corresponding smooth classification theory was studied. Let f,g: (R\({}^ n\times R^ r,0)\to (R^ p,0)\) be smooth map germs. f and g are P-V-equivalent if there exists a homeomorphism germ \(\Phi\) : (R\({}^ n\times R^ r,0)\to (R^ n\times R^ r,0)\) of the form \(\Phi (x,u)=(\phi^ 1(x,u),\phi (u))\) such that \(\Phi (f^{-1}(0))=g^{-1}(0)\). The main results of the paper are a genericity theorem and a classification theorem for map germs which are finitely determined with respect to P-V-equivalence. The genericity theorem shows that finite P-V-determinacy is, in a certain sense, a generic property for smooth map germs. The theorem is analogous to a theorem given in an article of \textit{A. du Plessis} [Topology 21, 131-156 (1982; Zbl 0499.58007)]. The classification theorem shows that for each integer \(\ell \geq 1\), there exists a semialgebraic set \(\Sigma_ 0^{\ell}(n+r,p)\subset J^{\ell}(n+r,p)\) such that \(\lim_{\ell \to \infty} co\dim \Sigma^ 1_ 0(n+r,p)=\infty\) and such that \(J^{\ell}(n+r,p)-\Sigma_ 0^{\ell}(n+r,p)\) has a finite, semialgebraic Whitney stratification which, among other things, has the property that map germs representing jets in the same stratum are P-V- equivalent.