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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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