Local topological properties of differentiable mappings. II.

*(English)*Zbl 0599.58010In the preceding paper [Invent. Math. 65, 227-250 (1981; Zbl 0499.58008)] it was shown that an almost every \(C^{\infty}\) mapgerm: \((R^ n,0)\to (R^ p,0)\), \(n\leq p\), has rather good topological structures. In particular it was shown that they are topologically equivalent to the cones of topologically stable mappings of \(S^{n-1}\) into \(S^{p-1}\), where the cone of a mapping \(f: X\to Y\) is the mapping Cf: \(X\times [0,1)/X\times \{0\}\to Y\times [0,1)/Y\times \{0\}\) defined by \(Cf(x,t)=(f(x),t).\)

This paper has two purposes. One is to show similar generic properties for the remaining case \(n>p\). The other is to show, as an application of these generic properties, that for almost every mapping into the plane f: (R\({}^ n,0)\to (R^ 2,0)\) a Poincaré-Hopf type equality, in some cases the Morse inequalities as well, holds between the Betti numbers of the set \(f^{-1}(0)\cap S_{\epsilon}^{n-1}\) and the indices of the singular points of f appearing around the origin, where \(S_{\epsilon}^{n-1}=\{x\in R^ n| \| x\| =\epsilon \}\) and \(\epsilon\) is supposed to be small. Here our Poincaré-Hopf type equality is as follows. Defining a function \(\theta\) : \(R^ 2-\{0\}\to (R\) mod \(2\pi)\) by \[ x+iy=\sqrt{x^ 2+y^ 2}e^{i\theta (x,y)},\quad (x,y)\in R^ 2-\{0\}, \] the composed mapping \(\theta\) \(\circ f: D^ n_{\epsilon}\cap f^{-1}(S^ 1_{\delta})\to (R\) mod \(2\pi)\) can be regarded as a Morse function. Although it is not a Morse function in the strict sense (its values are not in R but in R mod \(2\pi)\), we can define the indices of critical points of \(\theta\) \(\circ f: D^ n_{\epsilon}\cap f^{-1}(S^ 1_{\delta})\to (R\) mod \(2\pi)\) as usual. Now we set \(m_ i(f)= the\) number of critical points having index i of the Morse function \(\theta\) \(\circ f: D^ n_{\epsilon}\cap f^{- 1}(S^ 1_{\delta})\to R\) mod \(2\pi\), \(b_ i(M)= the\) ith Betti number of a manifold M, \(\chi (M)=\sum (-1)^ ib_ i(M)\) the Euler characteristic number of M. Theorem. For a generic map-germ f: (R\({}^ n,0)\to (R^ 2,0)\), the numbers \(m_ i(f)\) and \(b_ j(f^{-1}(0)\cap S_{\epsilon}^{n-1})\) are independent of \(\epsilon\) and \(\delta\) provided that \(\epsilon\) and \(\delta\) are sufficiently small, and we have the following Poincaré-Hopf type equality; \[ \sum^{n-1}_{i=0}(- 1)^ im_ i(f)+\chi (f^{-1}(0)\cap S_{\epsilon}^{n-1})=\chi (S^{n-1}). \]

This paper has two purposes. One is to show similar generic properties for the remaining case \(n>p\). The other is to show, as an application of these generic properties, that for almost every mapping into the plane f: (R\({}^ n,0)\to (R^ 2,0)\) a Poincaré-Hopf type equality, in some cases the Morse inequalities as well, holds between the Betti numbers of the set \(f^{-1}(0)\cap S_{\epsilon}^{n-1}\) and the indices of the singular points of f appearing around the origin, where \(S_{\epsilon}^{n-1}=\{x\in R^ n| \| x\| =\epsilon \}\) and \(\epsilon\) is supposed to be small. Here our Poincaré-Hopf type equality is as follows. Defining a function \(\theta\) : \(R^ 2-\{0\}\to (R\) mod \(2\pi)\) by \[ x+iy=\sqrt{x^ 2+y^ 2}e^{i\theta (x,y)},\quad (x,y)\in R^ 2-\{0\}, \] the composed mapping \(\theta\) \(\circ f: D^ n_{\epsilon}\cap f^{-1}(S^ 1_{\delta})\to (R\) mod \(2\pi)\) can be regarded as a Morse function. Although it is not a Morse function in the strict sense (its values are not in R but in R mod \(2\pi)\), we can define the indices of critical points of \(\theta\) \(\circ f: D^ n_{\epsilon}\cap f^{-1}(S^ 1_{\delta})\to (R\) mod \(2\pi)\) as usual. Now we set \(m_ i(f)= the\) number of critical points having index i of the Morse function \(\theta\) \(\circ f: D^ n_{\epsilon}\cap f^{- 1}(S^ 1_{\delta})\to R\) mod \(2\pi\), \(b_ i(M)= the\) ith Betti number of a manifold M, \(\chi (M)=\sum (-1)^ ib_ i(M)\) the Euler characteristic number of M. Theorem. For a generic map-germ f: (R\({}^ n,0)\to (R^ 2,0)\), the numbers \(m_ i(f)\) and \(b_ j(f^{-1}(0)\cap S_{\epsilon}^{n-1})\) are independent of \(\epsilon\) and \(\delta\) provided that \(\epsilon\) and \(\delta\) are sufficiently small, and we have the following Poincaré-Hopf type equality; \[ \sum^{n-1}_{i=0}(- 1)^ im_ i(f)+\chi (f^{-1}(0)\cap S_{\epsilon}^{n-1})=\chi (S^{n-1}). \]