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This book is primarily concerned with the study of cohomology theories of general topological spaces with "general coefficient systems. " Sheaves play several roles in this study. For example, they provide a suitable notion of "general coefficient systems. " Moreover, they furnish us with a common method of defining various cohomology theories and of comparison between different cohomology theories. The parts of the theory of sheaves covered here are those areas impor­ tant to algebraic topology. Sheaf theory is also important in other fields of mathematics, notably algebraic geometry, but that is outside the scope of the present book. Thus a more descriptive title for this book might have been Algebraic Topology from the Point of View of Sheaf Theory. Several innovations will be found in this book. Notably, the con­ cept of the "tautness" of a subspace (an adaptation of an analogous no­ tion of Spanier to sheaf-theoretic cohomology) is introduced and exploited throughout the book. The fact that sheaf-theoretic cohomology satisfies 1 the homotopy property is proved for general topological spaces. Also, relative cohomology is introduced into sheaf theory. Concerning relative cohomology, it should be noted that sheaf-theoretic cohomology is usually considered as a "single space" theory.

Inhaltsverzeichnis

Chapter I. Sheaves and Presheaves

Abstract
In this chapter we shall develop the basic properties of sheaves and presheaves and shall give many of the fundamental definitions to be used throughout the book. In Sections 2 and 5 various algebraic operations on sheaves are introduced. If we are given a map between two topological spaces, then a sheaf on either space induces, in a natural way, a sheaf on the other space, and this is the topic of Section 3. Sheaves on a fixed space form a category whose morphisms are called homomorphisms. In Section 4, this fact is extended to the collection of sheaves on all topological spaces with morphisms now being maps f of spaces together with so-called f-cohomomorphisms of sheaves on these spaces. In Section 6 the basic notion of a family of supports is defined and a fundamental theorem is proved concerning the relationship between a certain type of presheaf and the cross-sections of the associated sheaf. This theorem is applied in Section 7 to show how, in certain circumstances, the classical singular, Alexander-Spanier, and de Rham cohomology theories can be described in terms of sheaves.
Glen E. Bredon

Chapter II. Sheaf Cohomology

Abstract
In this chapter we shall define the sheaf-theoretic cohomology theory and shall develop many of its basic properties.
Glen E. Bredon

Chapter III. Comparison with Other Cohomology Theories

Abstract
We return in this chapter to the classical singular, Alexander-Spanier, de Rham, and Čech cohomology theories. It is shown that under suitable restrictions, these theories are equivalent to sheaf-theoretic cohomology. Homomorphisms induced by maps, cup products, and relative cohomology are also discussed at some length. In Section 3 the direct natural transformation between singular theory and de Rham theory, which is important in the applications, is considered.
Glen E. Bredon

Chapter IV. Applications of Spectral Sequences

Abstract
In this chapter we shall assume that the reader is familiar with the theory of spectral sequences, especially with the spectral sequences of double complexes. This basic knowledge is applied specifically to the theory of sheaves in Sections 1 and 2. See Appendix A for an outline of the parts of the theory of spectral sequences we shall need.
Glen E. Bredon

Chapter V. Borel-Moore Homology

Abstract
Throughout this chapter all spaces dealt with are assumed to be locally compact Hausdorff spaces. The base ring L will be taken to be a principal ideal domain, and all sheaves are assumed to be sheaves of L-modules. Note that over a principal ideal domain (and, more generally, over a Dedekind domain) a module is injective if and only if it is divisible.
Glen E. Bredon

Chapter VI. Cosheaves and Čech Homology

Abstract
In this short chapter we study the notion of cosheaves on general topological spaces and we go into it a bit deeper than was done in Chapter V. Our main purpose, in this chapter, is to obtain isomorphism criteria connecting various homology theories. With the minor exceptions of some definitions, and excepting the sections (10 and 11) concerning Borel-Moore homology, this chapter does not depend on Chapter V.
Glen E. Bredon

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