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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On a simplicial complex associated to the monodromy


Author: Gerald Leonard Gordon
Journal: Trans. Amer. Math. Soc. 261 (1980), 93-101
MSC: Primary 32C40; Secondary 14D05, 32G13
MathSciNet review: 576865
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Abstract: Suppose we have a complex analytic family, $ {V_t}$, $ \left\vert t \right\vert\, \leqslant \,1$, such that the generic fibre is a nonsingular complex manifold of complex dimension n. Let T denote the monodromy induced from going once around the singular fibre and let I denote the identity map. We shall associate to the singular fibre a simplicial complex $ \Gamma $, which is at most n-dimensional. Then under certain conditions on the family $ {V_t}$ (which are satisfied for the Milnor fibration of an isolated singularity or if the $ {V_t}$ are compact Kähler), there is an integer $ N\, > \,0$ such that $ {({T^N}\, - \,I)^k}{H_k}({V_t})\, = \,0$ if and only if $ {H_k}(\Gamma )\, = \,0$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1980-0576865-1
PII: S 0002-9947(1980)0576865-1
Article copyright: © Copyright 1980 American Mathematical Society