Lectures on Diffeology

This book is a set of lecture notes following the lectures given à Shantou University in 2020/21. They are aimed at students interested in differential geometry, especially in directions not ordinarily covered by the classical theory.
These notes will be published by Beijing World Publishing Corp. But until then, I leave the pdf for free download. Any comment is welcome.
They are divided into two parts: firstly, a series of lectures, which were presented as part of a special program at Shantou University. They summarize the major chap- ters of diffeology: axiomatics, categories, homotopy, Cartan calculus, fiber bundles, etc. They serve as an introduction to the various sections of diffeology in the cor- responding chapters of the “Diffeology” textbook, which remains the formal basis for a course in diffeology. They do not replace the textbook, but serve as a kind of springboard. The second part consists of a series of notes, remarks or exercises, chosen in particular because they are not covered by standard geometry of manifolds. They show, for example, how diffeology deals specifically with infinite-dimensional spaces or singular quotients, or both. We construct in particular a symplectic diffeological space without Hamiltonian diffeomorphisms. We show how to deal with the space of geodesics of the torus, despite the fact that it is far from being a manifold. We present also the axiomatics of a Riemannian diffeology program, which is not cov- ered in the textbook. We compute the variation of the holonomy of a generalized torus fiber bundle. We develop the diffeology of Fourier coefficients appearing in the symplectic structure of the set of complex periodic functions, and we apply a reduction process to a singular Hamiltonian level to show how diffeology deals with singularities in infinite-dimensional systems. That constructs this way a clas- sifying space for finite dimensional quasi-spheres. All these examples are chosen to familiarize the student with certain techniques in the versatility environment of diffeology.
At the end, we discuss how diffeology can be seen as the perfect framework for differential geometry, to the point where it is exactly what we expect of differential geometry; with all the flexibility and generality a geometer can dream of, without each time inventing a heuristic framework that momentarily satisfies his needs.
To support our point of view, we take the example of the action of diffeomorphisms of the square to illustrate Klein’s stratification, which exists on any diffeological space, and which embodies its internal geometry, in the sense of Felix Klein. Differential geometry thus becomes the geometry of the groups of diffeomorphisms.

Lectures on Diffeology, the book