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Generalized local and global Sebastiani-Thom type theorems. (English) Zbl 0744.32020
Let $$g: (\mathbb{C}^ n,0)\to(\mathbb{C},0)$$ resp. $$h: (\mathbb{C}^ m,0)\to(\mathbb{C},0)$$ be analytic germs with Milnor fiber $$G$$ resp. $$H$$. Let $$p: (\mathbb{C}^ 2,0)\to(\mathbb{C},0)$$ 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 $$\mu_ p=1-\chi_ p$$ circles. We define $$n_ c=0$$ if $$c$$ is a factor of $$p$$ and $$n_ c$$= the intersection multiplicity $$m_ 0(p,c)$$ otherwise. Symmetrically we define $$n_ d$$. Then $$n_ c$$ is the number of points of the intersection $$P\cap\{c=0\}$$.
Theorem. The Milnor fiber $$F$$ of the analytic germ $$f=p(h,g): (\mathbb{C}^ n\times\mathbb{C}^ m,0)\to(\mathbb{C},0)$$ 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 $$G\times H$$ $$\hbox{n}_ c$$ copies of $$\hbox{Con} G\times H$$ and $$\hbox{n}_ d$$ copies of $$G\times \hbox{Con}H$$.
Theorem. The zeta function of $$f$$ is determined by $\zeta_ f(\lambda)=\zeta_ g(\lambda^{n_ d})\cdot\zeta_ h(\lambda^{n_ c})\cdot\prod_ g\det \Delta(\lambda^{m_ 1}E_{q,1},\lambda^{m_ 2}E_{q,2},\lambda^{m_ 3}\cdot I,\dots,\lambda^{m_ r}\cdot I)^{(-1)^ g}.$

MSC:
 32S50 Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants
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References:
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