ABSTRACT
The topology of complex isolated hypersurface singularities has been long studied by many authors, as for instance Milnor [3], Lê Dũng Tráng [2] and many more. There is a beautiful and well developed theory to this respect, though there are still many things to be understood. This note is concerned with the real counterpart of this theory and it was inspired by [3, 4]. The motivation was to consider a class of real hypersurface singularities that arise naturally from complex singularities. We study the topology of the real hypersurfaces that arise when we consider a ℂ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203912089/10e95e2c-eca6-48d5-adf5-58298374b4f7/content/eq2575.tif"/> -valued holomorphic function H on an open set U https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203912089/10e95e2c-eca6-48d5-adf5-58298374b4f7/content/eq2576.tif"/> in ℂ n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203912089/10e95e2c-eca6-48d5-adf5-58298374b4f7/content/eq2577.tif"/> and we compose this function with the projection onto a real line through the origin in ℂ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203912089/10e95e2c-eca6-48d5-adf5-58298374b4f7/content/eq2578.tif"/> . We show that, for a fixed function H, all these hypersurfaces are homeomorphic and the link of the singularity is the double of the Milnor fibre of the holomorphic function H. It would be interesting to make a similar study for isolated complete intersection complex singularities with codimension higher than 1. This general case is certainly more interesting, and more difficult, than the one envisaged in this note, since the type of the real singularities so obtained, by composing the complex map with the projection onto a real line, will vary as the corresponding line intersects the discriminant of the original function. So this article should be regarded as the first approach to that general, and hopefully interesting, problem.
