Sheaf TheorySpringer Science & Business Media, 24 janv. 1997 - 504 pages 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. |
Table des matières
Sheaf Cohomology | 33 |
4 | 42 |
6 | 49 |
10 | 66 |
11 | 75 |
13 | 91 |
14 | 100 |
15 | 106 |
11 | 257 |
13 | 264 |
BorelMoore Homology | 279 |
13 | 352 |
14 | 364 |
17 | 374 |
19 | 403 |
5 | 426 |
17 | 115 |
18 | 133 |
21 | 154 |
Applications of Spectral Sequences | 179 |
3 | 203 |
4 | 211 |
Extension of a support family by a family on the base space | 219 |
10 | 249 |
8 | 434 |
11 | 442 |
A Spectral Sequences | 449 |
Solutions to Selected Exercises | 455 |
Bibliography | 487 |
List of Selected Facts | 493 |
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Expressions et termes fréquents
acyclic assume c-soft canonical map Čech cohomology chain map coefficient sequence cohomology groups cohomology sequence commutative diagram compact Hausdorff space compact spaces connected constant sheaf Corollary cosheaf countable cup product defined definition denote differential sheaf dimz direct limits exact functor exact sequence example Exercise f-cohomomorphism family of supports flabby follows H¹(X Hausdorff space hence homology homomorphism homotopy Hp(X induces injective resolution integer L-module Lemma Leray sheaf Leray spectral sequence Let f locally closed locally compact Hausdorff locally constant locally finite monomorphism natural isomorphism notation Note open neighborhood open sets paracompact paracompactifying family phism presheaf principal ideal domain Proof Proposition restriction result ring with unit sheaves singular cohomology stalks subsets subsheaf Suppose surjective taut Theorem theory topology torsion-free U₁ whence zero