Journal of the London Mathematical Society



THE DENEF–LOESER ZETA FUNCTION IS NOT A TOPOLOGICAL INVARIANT


E. ARTAL BARTOLO a1 1 , P. CASSOU-NOGUÈS a2, I. LUENGO a3 1 and A. MELLE HERNÁNDEZ a3 1
a1 Departamento de Matemáticas, Universidad de Zaragoza, Campus Plaza San Francisco s/n, E-50009 Zaragoza, Spain; artal@posta.unizar.es
a2 Laboratoire de Mathématiques Pures, Université Bordeaux I, 350 Cours de la Libération, 33405 Talence Cedex, France; cassou@math.u-bordeaux.fr
a3 Departamento de Álgebra, Universidad Complutense, Ciudad Universitaria s/n, E-28040 Madrid, Spain; iluengo@eucmos.sim.ucm.es, amelle@eucmos.sim.ucm.es

Abstract

An example is given which shows that the Denef–Loeser zeta function (usually called the topological zeta function) associated to a germ of a complex hypersurface singularity is not a topological invariant of the singularity. The idea is the following. Consider two germs of plane curves singularities with the same integral Seifert form but with different topological type and which have different topological zeta functions. Make a double suspension of these singularities (consider them in a 4-dimensional complex space). A theorem of M. Kervaire and J. Levine states that the topological type of these new hypersurface singularities is characterized by their integral Seifert form. Moreover the Seifert form of a suspension is equal (up to sign) to the original Seifert form. Hence these new singularities have the same topological type. By means of a double suspension formula the Denef–Loeser zeta functions are computed for the two 3-dimensional singularities and it is verified that they are not equal.

(Received April 24 2001)



Footnotes

1 The first author is partially supported by DGES PB97-0284-C02-02; the third and fourth authors are partially supported by DGES PB97-0284-C02-01.