Manifolds, sheaves, and cohomology (Q2356592)

The main aim of this book is to introduce powerful techniques used in modern Algebraic and Differential Geometry, fundamentally focusing on the relation between local and global properties of geometric objects and on the obstructions to passing from the former to the latter. The objects in question are real and complex manifolds, viewed as examples of ringed spaces (topological spaces together with a sheaf of commutative algebras over some fixed, usually commutative, ring.) This approach requires that one first develops a theory of sheaves, which are introduced here at length in Chapter 3, together with the equivalent notion of étale spaces. The properties of real and complex manifolds are then mostly achieved through a study of the cohomology of sheaves and of complexes of sheaves, starting with first Čech cohomology in Chapter 7, and concluding in Chapter 10 with higher-degree cohomology and its application to the de Rham cohomology of manifolds. The book can be divided into three parts: first, in Chapters 1 through 3, the notions of presheaf and sheaf are presented (after some preliminaries on topological spaces and algebraic topology), and their properties explored; then, in Chapters 4 to 6 and 8, premanifolds and manifolds are introduced, including the construction of some of these objects from others and the definition of their tangent spaces, and Lie groups and bundles are studied; finally, in Chapter 7 and Chapters 9 through 11, the cohomology of sheaves and of complexes of sheaves is explored, and some local-global results obtained from it. The readership for this book will mostly consist of beginner to intermediate graduate students, and it may serve as the basis for a one-semester course on the cohomology of sheaves and its relation to real and complex manifolds. The prerequisites needed to follow the arguments are mostly a good grasp on General Topology, some exposure to Algebraic Topology (involving such topics as the definition of first homotopy groups, covering spaces, and diagram chases in exact sequences), Category Theory, and Commutative Algebra. All of these topics are reviewed in the excellent appendixes at the end of the book. Each chapter also features a list of exercises that in some cases feature important examples of the definitions presented before. If one wants simply to skim through this book, the suggestion would be to focus on Chapters 3 (on sheaves), 4 (on manifolds), and then go on to Chapters 7 (on first Čech cohomology), 9 (on soft sheaves and their relation to partitions of unity) and 10 (on the cohomology of complexes of sheaves), and turn to the other chapters whenever needed. The novelty of this text, besides the self-contained compilation of topics that are usually encountered in several distinct textbooks, is the non-classical approach to the definition of real and complex premanifolds and manifolds. This is done in Chapter 4, where real premanifolds appear as locally \(\mathbb{R}\)-ringed spaces that are locally isomorphic to the sheaf of \(\mathbb{R}\)-valued \(C^\alpha\)-functions (for some fixed \(\alpha\)) and complex premanifolds are locally \(\mathbb{C}\)-ringed spaces locally isomorphic to the sheaf of holomorphic functions. The advantage of this non-atlas-based approach is that, when one is up to a certain point permitted to use differentiable functions with domain defined only ``up to an unspecified open neighbourhood'', it becomes easier to write down and prove some properties of premanifolds and manifolds and, moreover, some of these properties can be viewed with a higher degree of generality. The classical definition is also presented later in the same chapter, and there is no great risk here of confusion to unexperienced readers, as most will have encountered the classical notions previously. For established researchers in the areas covered, there will not be anything substantially new here, but this book is nonetheless a good source of standard technical results that are mostly proved at length. Some of these are not elaborated upon, e.g. the Poincaré lemma in Chapter 8. Crucial extensions of some of the definitions to more general settings are also beyond the scope of the text, but are hinted upon, for example the notion (in Chapter 10) of the cohomology of a complex of sheaves as the right derivation of a left exact functor between abelian categories.