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The Lê numbers of the square of a function and their applications
Author(s) -
Fernández de Bobadilla Javier,
Gaffney Terence
Publication year - 2008
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdm101
Subject(s) - hypersurface , gravitational singularity , mathematics , singularity , square (algebra) , pure mathematics , euler's formula , invariant (physics) , generalization , function (biology) , topology (electrical circuits) , mathematical analysis , combinatorics , geometry , mathematical physics , evolutionary biology , biology
Lê numbers were introduced by Massey with the purpose of numerically controlling the topological properties of families of non‐isolated hypersurface singularities and describing the topology associated with a function with non‐isolated singularities. They are a generalization of the Milnor number for isolated hypersurface singularities. In this note the authors investigate the composite of an arbitrary square‐free f and z 2 . They get a formula for the Lê numbers of the composite, and consider two applications of these numbers. The first application is concerned with the extent to which the Lê numbers are invariant in a family of functions which satisfy some equisingularity condition, the second is a quick proof of a new formula for the Euler obstruction of a hypersurface singularity. Several examples are computed using this formula including any X defined by a function which only has transverse D ( q, p ) singularities off the origin.