Sheaf Theory [[electronic resource] /] / by Glen E. Bredon

• Autore: Bredon Glen E
• Pubblicazione: New York, NY : , : Springer New York : , : Imprint : Springer, , 1997
• Edizione: 2nd ed. 1997.
• Descrizione fisica: 1 online resource (XI, 504 p.)
• Disciplina: 512
• Soggetto topico: Algebra; Algebraic topology; Algebraic Topology
• Note generali: Bibliographic Level Mode of Issuance: Monograph
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: I Sheaves and Presheaves -- Definitions -- 2 Homomorphisms, subsheaves, and quotient sheaves -- 3 Direct and inverse images -- 4 Cohomomorphisms -- 5 Algebraic constructions -- 6 Supports -- 7 Classical cohomology theories -- Exercises -- II Sheaf Cohomology -- 1 Differential sheaves and resolutions -- 2 The canonical resolution and sheaf cohomology -- 3 Injective sheaves -- 4 Acyclic sheaves -- 5 Flabby sheaves -- 6 Connected sequences of functors -- 7 Axioms for cohomology and the cup product -- 8 Maps of spaces -- 9 ?-soft and ?-fine sheaves -- 10 Subspaces -- 11 The Vietoris mapping theorem and homotopy invariance -- 12 Relative cohomology -- 13 Mayer-Vietoris theorems -- 14 Continuity -- 15 The Künneth and universal coefficient theorems -- 16 Dimension -- 17 Local connectivity -- 18 Change of supports; local cohomology groups -- 19 The transfer homomorphism and the Smith sequences -- 20 Steenrod’s cyclic reduced powers -- 21 The Steenrod operations -- Exercises -- III Comparison with Other Cohomology Theories -- 1 Singular cohomology -- 2 Alexander-Spanier cohomology -- 3 de Rham cohomology -- 4 ?ech cohomology -- Exercises -- IV Applications of Spectral Sequences -- 1 The spectral sequence of a differential sheaf -- 2 The fundamental theorems of sheaves -- 3 Direct image relative to a support family -- 4 The Leray sheaf -- 5 Extension of a support family by a family on the base space -- 6 The Leray spectral sequence of a map -- 7 Fiber bundles -- 8 Dimension -- 9 The spectral sequences of Borel and Cartan -- 10 Characteristic classes -- 11 The spectral sequence of a filtered differential sheaf -- 12 The Fary spectral sequence -- 13 Sphere bundles with singularities -- 14 The Oliver transfer and the Conner conjecture -- Exercises -- V Borel-Moore Homology -- 1 Cosheaves -- 2 The dual of a differential cosheaf -- 3 Homology theory -- 4 Maps of spaces -- 5 Subspaces and relative homology -- 6 The Vietoris theorem, homotopy, and covering spaces -- 7 The homology sheaf of a map -- 8 The basic spectral sequences -- 9 Poincaré duality -- 10 The cap product -- 11 Intersection theory -- 12 Uniqueness theorems -- 31 Uniqueness theorems for maps and relative homology -- 14 The Künneth formula -- 15 Change of rings -- 16 Generalized manifolds -- 17 Locally homogeneous spaces -- 18 Homological fibrations and p-adic transformation groups -- 19 The transfer homomorphism in homology -- 20 Smith theory in homology -- Exercises -- VI Cosheaves and ?ech Homology -- 1 Theory of cosheaves -- 2 Local triviality -- 3 Local isomorphisms -- 4 Cech homology -- 5 The reflector -- 6 Spectral sequences -- 7 Coresolutions -- 8 Relative ?ech homology -- 9 Locally paracompact spaces -- 10 Borel-Moore homology -- 11 Modified Borel-Moore homology -- 12 Singular homology -- 13 Acyclic coverings -- 14 Applications to maps -- Exercises -- A Spectral Sequences -- 1 The spectral sequence of a filtered complex -- 2 Double complexes -- 3 Products -- 4 Homomorphisms -- B Solutions to Selected Exercises -- Solutions for Chapter I -- Solutions for Chapter II -- Solutions for Chapter III -- Solutions for Chapter IV -- Solutions for Chapter V -- Solutions for Chapter VI -- List of Symbols -- List of Selected Facts.
• Sommario/riassunto: 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 impor­tant 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 con­cept of the "tautness" of a subspace (an adaptation of an analogous no­tion 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.
• Titolo autorizzato: Sheaf Theory
• ISBN: 1-4612-0647-2
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910789350003321
• Lo trovi qui: Univ. Federico II
• Serie: Graduate Texts in Mathematics, . 0072-5285 ; ; 170