By Otto Mutzbauer

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**Extra resources for Analytische Geometrie**

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Hamm f) H) e) We look at gjB \ X \ ff D sg. 2 there are no critical points for the restriction to the boundary (above Dˇ ). Dˇ //: e) H) f) Assume that f) is wrong. X n Y / \ BV . Dˇ / ! DP ˛ Dˇ is proper, so the image is analytic of dimension 1, because of Sard’s theorem: D 1. The image curve must intersect s D const, where 0 < jsj ˛. So we have that gjB \ X \ ff D sg has critical points. These cause a difference between the Euler characteristics considered in e), contradiction. 3. This follows from the preceding theorem: a) H) b).

F0g/; dim Y D n 1: We assume that X n Y is smooth. As in Section 2, let j W X n Y ! X be the inclusion. z1 ; : : : ; zn ; 0/k Ä g and S D @B : (In this and the next section we could also take the usual ball, resp. 1. 1. X; Y /. Assume that the hyperplane fg D 0g intersects all strata of X transversally within some punctured neighborhood of 0. f; g/ W B \ X \ f defines a C 1 fibre bundle. Dˇ / ! DP ˛ Dˇ 26 Helmut A. Hamm We want to weaken the hypothesis of this theorem by assuming that ˆg j CXnY is acyclic in some punctured neighborhood of 0.

G CX /0 is acyclic is fulfilled in particular when X D CmC1 (with n D m C 1, of course). 1. 3. Example. 3 does not hold. 2 is applicable. t is constant. 3. z1 ; : : : ; zn ; 0/k: Let † be the critical set of . ; Re g/jY \ fIm g D 0g n Z. After shrinking U if necessary, each branch of the closure of † is parametrized by a real analytic curve 7! /, with . / D 0, and we may choose the parametrisation in such a way that . // D ˙ Ä . In this way we see that there is k > 0 such that along † we have the inequality jRe gjk < < jRe gj1=k .

### Analytische Geometrie by Otto Mutzbauer

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