Claudio Gorodski - Smooth Manifolds

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The concept of smooth manifold is ubiquitous in Mathematics. Indeed smooth manifolds appear as Riemannian manifolds in differential geometry, spacetimes in general relativity, phase spaces and energy levels in mechanics, domains of definition of ODEs in dynamical systems, Riemann surfaces in the theory of complex analytic functions, Lie groups in algebra and geometry, to name a few instances. The notion took some time to evolve until it reached its present form in H. Whitneys celebrated Annals of Mathematics paper in 1936 [Whi36]. Whitneys paper in fact represents a culmination of diverse historical developments which took place separately, each in a different domain, all striving to make the passage from the local to the global.

From the modern point of view, the initial goal of introducing smooth manifolds is to generalize the methods and results of differential and integral calculus, in special, the inverse and implicit mapping theorems, the theorem on existence, uniqueness, and regularity of ODEs and Stokes theorem. As usual in Mathematics, once introduced such objects start to attract interest on their own and new structure is uncovered. The subject of Differential Topology studies smooth manifolds per se. Many important results about the topology of smooth manifolds were obtained in the 1950s and 1960s in the high dimensional range. For instance, there exist topological manifolds admitting several non-diffeomorphic smooth structures (Milnor [Mil56], in the case of S7), and there exist topological manifolds admitting no smooth structure at all (Kervaire [Ker60]). Moreover, the generalized Poincar conjecture (GPC: Is every compact n-manifold with the homotopy type of the n-sphere equivalent to the n-sphere?) was proved in dimensions bigger than 4, with increasing generality of assumptions about the manifold (smooth, combinatorial, topological), by Smale, Stallings, Zeeman, and Newman in the 1960s. Major developments in dimension 4 took place in the 1980s: Freedman gave a complete classification of simply-connected compact topological 4-manifolds, including a proof of GPC; and Donaldson provided major contributions to the question of determining which topological 4-manifolds admit smooth structures. Despite these breakthroughs, the classification of smooth structures on smoothable topological 4-manifolds remains largely terra incognita. In lower dimensions, every topological manifold is smoothable. The very important case of dimension 3 has seen tremendous development after the geometrization program of Thurston (late 1970s), Hamiltons Ricci flow (1981), and Perelmans Ricci flow with surgery (2003), including a proof of GPC by Perelman, and continues toattract a lot of attention. We mention in closing that the topology of compact surfaces is a classical subject already tackled in the nineteenth century [GX13, App. D].

The aim of these notes is much more modest. They are intended as a beginning graduate-level textbook and, in principle, presume a reasonable understanding of calculus (including the proofs of the inverse and implicit mapping theorems and the formula of change of variables in multiple integrals, roughly corresponding to most of the contents in the first three chapters of [Spi65]), the statement of the existence and uniqueness theorems of ordinary differential equations as in [Hur70, Chap. 2, 5], basic linear algebra and some point set topology; some experience with fundamental groups and covering spaces is also of help. More important than the details about specific prerequisites, that students may or may not have, is the level of maturity that one has acquired by working with key examples in those mathematical theories. The text is also suitable for a supervised reading course. On the other hand, on account of the most prominent concise character of these notes, we hope that their style will stimulate students to develop perseverance in case they are used for independent study.