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The Vanishing Topology of Non Isolated Singularities

By Dirk Siersma


this paper we want to discuss especially non-isolated singularities. Although the properties (e.g. the topological type of the Milnor fibre) seem to be more complicated than for isolated singularities, there is a lot of interesting structure available. Let \Sigma = \Sigma(f ) be the singular locus of f: For every point of \Sigma we can do the Milnor construction. So for every x 2 \Sigma we have a (local) Milnor fibration (e.g. a space E (x) , the local Milnor fibre F (x) and a Milnor monodromy T (x) ). We want to investigate the relation 1 between these objects for all x 2 \Sigma. This is (near to) the study of the sheaf of vanishing cycles [De]. To be more precise: one could try to define a stratification of \Sigma in such a way that two points of \Sigma are in the same stratum if they can joined by a (continuous) path, such that there exits a (continuous) family of Milnor fibrations of constant fibration type. According to a result of Massey, the constancy of Le numbers (for definition see the contribution of Gaffney [Ga] in this Volume) implies constancy of the fibration type under certain dimension conditions (more precisely s n \Gamma 2 for the homotopy-type and s n \Gamma 3 for the diffeomorphism-type). We refer to Massey's monograph [Ma-2] for details and for many other related facts. Let us suppose that we end op with a situation, where we have stratified \Sigma (according to the above principle): \Sigma = \Sigm

Year: 2000
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