Transversality, currents and Morse theory. A course of differential topology. Exercises proposed by François Labourie (Q2856001)

This short book is a course on Differential Topology, which is mainly oriented to introducing Morse theory, but covering an amazing number of topics. The first chapter studies the fundamental notions of manifold, tangent bundle, partition of unity, tubular neighborhood and fiber bundle. The second chapter is devoted to differential forms, their exterior product, the coboundary operator, the de~Rham cohomology, integration of differential forms, the Stokes formula, the Poincaré lemma, and the operator on differential forms defined by a smooth homotopy. The third chapter deals with vector fields, their flows and bracket product, the Lie derivative, the Cartan formula and other basic equalities, the divergence, the Morse lemma, the Darboux lemma for symplectic forms, the Frobenius theorem, foliations, the Godbillon-Vey class, and Lie groups.NEWLINENEWLINEThe fourth chapter studies de~Rham currents (in particular, currents defined by differential forms and oriented submanifolds), the boundary operator on currents, the corresponding homology, regularization of currents, and the isomorphism between de~Rham cohomology and the homology of the complex of currents. The fifth chapter is devoted to transversality, the Sard theorem, Morse functions, transversality for families of maps, the \(C^\infty\) topology, the transversality theorem, which is used to prove the Whitney theorems of immersion and embedding, and the theorem of Thom stating that the subspace of Morse functions on a closed manifold is open and dense with the \(C^\infty\) topology.NEWLINENEWLINEThe sixth chapter is devoted to Morse theory; it includes the study of gradients adapted to a Morse function, complete Riemannian metrics, the condition of Palais-Smale, sub-level sets, the mini-max principle for differential forms, models of a Morse function around critical points, the inequality of Ljusternik-Snirel'man, stable and unstable manifolds of the critical points, their interpretation as currents and the formula for their boundary (giving rise to the complex of Thom-Smale), and the fundamental theorem stating the isomorphism between the de~Rham cohomology and the cohomology of the Thom-Smale complex.NEWLINENEWLINEThis fundamental theorem is used to obtain the finite dimension of the de~Rham cohomology, the Poincaré duality, the Morse inequalities, the de~Rham theorem (stating the duality between homology and cohomology), and the Künneth formula.NEWLINENEWLINEThis final chapter also studies the Heegaard decomposition of closed \(3\)-manifolds, whose existence is shown by using Morse functions, and an appendix with technical results about the closure of the stable manifolds. Finally, the book contains a list of interesting exercises. All of these concepts are explained in a simple way, leaving some steps of the proofs for the readers or omitting some proofs (giving appropriate references instead), but keeping interesting observations. Thus it gives a nice panoramic view of the subject, which is very appropriate for a master course with a small number of hours. A small criticism is that it contains a few typographic errors that should be corrected if there are new editions.