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Zbl 0744.32020
Némethi, András
Generalized local and global Sebastiani-Thom type theorems. (English)
[J] Compos. Math. 80, No.1, 1-14 (1991). ISSN 0010-437X; ISSN 1570-5846/e

Let $g: (\bbfC\sp n,0)\to(\bbfC,0)$ resp. $h: (\bbfC\sp m,0)\to(\bbfC,0)$ be analytic germs with Milnor fiber $G$ resp. $H$. Let $p: (\bbfC\sp 2,0)\to(\bbfC,0)$ be an analytic germ in two variables ($c$ and $d$) with Milnor fiber $P$. In this paper, we suppose that $P$ is connected. In this case $P$ has the homotopy type of a bouquet of $\mu\sb p=1-\chi\sb p$ circles. We define $n\sb c=0$ if $c$ is a factor of $p$ and $n\sb c$= the intersection multiplicity $m\sb 0(p,c)$ otherwise. Symmetrically we define $n\sb d$. Then $n\sb c$ is the number of points of the intersection $P\cap\{c=0\}$.\par Theorem. The Milnor fiber $F$ of the analytic germ $f=p(h,g): (\bbfC\sp n\times\bbfC\sp m,0)\to(\bbfC,0)$ defined by $f(x,y)=p(g(x),h(y))$ has the homotopy type of a space obtained from the total space of a fiber bundle with base space $P$ and fiber $G\times H$ by gluing with the natural applications to a fiber $G\times H$ $\hbox{n}\sb c$ copies of $\hbox{Con} G\times H$ and $\hbox{n}\sb d$ copies of $G\times \hbox{Con}H$.\par Theorem. The zeta function of $f$ is determined by $$\zeta\sb f(\lambda)=\zeta\sb g(\lambda\sp{n\sb d})\cdot\zeta\sb h(\lambda\sp{n\sb c})\cdot\prod\sb g\det \Delta(\lambda\sp{m\sb 1}E\sb{q,1},\lambda\sp{m\sb 2}E\sb{q,2},\lambda\sp{m\sb 3}\cdot I,\dots,\lambda\sp{m\sb r}\cdot I)\sp{(-1)\sp g}.$$
[S.Ohyanagi (Kamahura)]

MSC 2000:: *32S50 Topological aspects with respect to singularities on analytic spaces

Keywords: Milnor fibering; Milnor number; homotopy type; zeta function

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