The first monograph on singularities of mappings for many years, this book provides an introduction to the subject and an account of recent developments concerning the local structure of complex analytic mappings.
Part I of the book develops the now classical real C and complex analytic theories jointly. Standard topics such as stability, deformation theory and finite determinacy, are covered in this part. In Part II of the book, the authors focus on the complex case. The treatment is centred around the idea of the "nearby stable object" associated to an unstable map-germ, which includes in particular the images and discriminants of stable perturbations of unstable singularities. This part includes recent research results, bringing the reader up to date on the topic.
By focusing on singularities of mappings, rather than spaces, this book provides a necessary addition to the literature. Many examples and exercises, as well as appendices on background material, make it an invaluable guide for graduate students and a key reference for researchers. A number of graduate level courses on singularities of mappings could be based on the material it contains.
Singularities of Mappings: The Local Behaviour of Smooth and Complex Analytic Mappings
Introduction.- Part I Thom-Mather Theory: Right-Left Equivalence, Stability,Versal Unfoldings, Finite Determinacy.- Manifolds and Smooth Mappings.- Left-Right Equivalence and Stability.- Contact Equivalence.- Versal Unfoldings.- Finite Determinacy.- Classification of Stable Germs by Their Local Algebras.- Part II Images and Discriminants: The Topology of Stable Perturbations.- Stable Images and Discriminants.- Multiple Points.- Calculating the Homology of the Image.- Multiple Points in the Target: The Case of Parameterised Hypersurfaces.- Appendix A: Jet Spaces and Jet Bundles.- Appendix B Stratifications.- Appendix C Background in Commutative Algebra.- Appendix D Local Analytic Geometry.- Appendix E Sheaves. References.- Index