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Topological series of isolated plane curve singularities. (English) Zbl 0708.57011

The author observes that although singularities occur in series and the simplest one have been given names by Arnol’d, it is not clear how to define what is meant by a series. A topological definition of series, for plane curve singularities, is proposed in this work: Let f be an element of the ring of convergent power series \(C\{\) x,y\(\}\) with a non-isolated singularity. The topological series belonging to f consists of all topological types of isolated singularities whose links arise as the splice of the link of f with some other link. (The splice operation is due to Siebenmann and in this case implies that “the Milnor fibration of an element of the series differs from that of f only in small neighborhoods of the components with higher multiplicities”.) The motivation for the definition, examples, and calculations of some topological invariants are also given, in a very readable and accessible style.
Reviewer: J.G.Timourian

MSC:

57R45 Singularities of differentiable mappings in differential topology
32S55 Milnor fibration; relations with knot theory
57R99 Differential topology
14H20 Singularities of curves, local rings
14B05 Singularities in algebraic geometry
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