Martinet J. Singularities of Smooth Functions and Maps

Cambridge: Cambridge University Press, 1982. — 271 p.These notes arose from a seminar held at the University of Michigan during the winter 1973-74, and a course given at the Pontificia Universidade Catolica do Rio de Janeiro (P.U.C.) from March to June 1974.
My aim was to present a detailed study of the most important features of the singularity theory for differentiable mappings, as it had developed in the fundamental work of H. Whitney, R. Thorn, J.N. Mather and some other mathematicians.
This subject is so rich that one has to make careful choices to give a coherent and significant idea of the theory. In this text, I have made the following choices£
Only the local theory is developed, that is, the study of germs of functions or mappings; but the transversality theorem, which is actually a local result, is included.
A prominent role is given to the notion of an unfolding, which I consider one of the most important concepts in this theory, technically as well as conceptually.
I highlight the differen~es and analogies between the singularity theory of functions (lRn ~ IR; only the group of local diffeornorphisrns of lRn is involved), and of mappings (lRP -+ lRq; both groups of local diffeomorphisms of 1Rp and IRq are involved).
A proof of the division theorem for differentiable functions is included; it is the only part of the theory where analysis is needed, but it is fundamental. I choose Lojasiewicz's beautiful proof, which is a pleasure to develop; moreover, it represents an introduction to some important techniques in differential analysis.