## Local topological properties of differentiable mappings. II.(English)Zbl 0599.58010

In 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; $\sum^{n-1}_{i=0}(- 1)^ im_ i(f)+\chi (f^{-1}(0)\cap S_{\epsilon}^{n-1})=\chi (S^{n-1}).$

### MSC:

 58C25 Differentiable maps on manifolds 58K99 Theory of singularities and catastrophe theory 57R45 Singularities of differentiable mappings in differential topology 58A35 Stratified sets

Zbl 0499.58008
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