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Journal of the London Mathematical Society Advance Access originally published online on January 31, 2008
Journal of the London Mathematical Society 2008 77(3):545-557; doi:10.1112/jlms/jdm101
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© 2008 London Mathematical Society

The Lê numbers of the square of a function and their applications

Javier Fernández de Bobadilla

C.S.I.C.
Spain

Terence Gaffney

Northeastern University
360 Huntington Avenue
Boston
MA 02115–5000
USA
gaff@neu.edu

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 z2. 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.

2000 Mathematics Subject Classification 14B05, 32S15, 32S50 (primary).

Received March 22, 2006; accepted March 6, 2007; published online January 31, 2008.


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