Differential Forms in Algebraic Topology

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Springer Science & Business Media, 17 abr 2013 - 338 páginas
The guiding principle in this book is to use differential forms as an aid in exploring some of the less digestible aspects of algebraic topology. Accord ingly, we move primarily in the realm of smooth manifolds and use the de Rham theory as a prototype of all of cohomology. For applications to homotopy theory we also discuss by way of analogy cohomology with arbitrary coefficients. Although we have in mind an audience with prior exposure to algebraic or differential topology, for the most part a good knowledge of linear algebra, advanced calculus, and point-set topology should suffice. Some acquaintance with manifolds, simplicial complexes, singular homology and cohomology, and homotopy groups is helpful, but not really necessary. Within the text itself we have stated with care the more advanced results that are needed, so that a mathematically mature reader who accepts these background materials on faith should be able to read the entire book with the minimal prerequisites. There are more materials here than can be reasonably covered in a one-semester course. Certain sections may be omitted at first reading with out loss of continuity. We have indicated these in the schematic diagram that follows. This book is not intended to be foundational; rather, it is only meant to open some of the doors to the formidable edifice of modern algebraic topology. We offer it in the hope that such an informal account of the subject at a semi-introductory level fills a gap in the literature.
 

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Table of Contents Preface Interdependence of the Sections Introduction
de Rham Theory 1 The de Rham Complex on ℝ
2 The MayerVietoris Sequence
3 Orientation and Integration
4 Poincaré Lemmas 5 The MayerVietoris Argument
6 The Thom Isomorphism
7 The Nonorientable Case
ChapterII The Čechde Rham Complex 8 The Generalized MayerVietoris Principle 9 More Examples and Applications of the MayerVietoris Principle
15 Cohomology with Integer Coefficients
16 The Path Fibration
17 Review of Homotopy Theory
18 Applications to Homotopy Theory
19 Rational Homotopy Theory
Characteristic Classes
20 Chern Classes of a Complex Vector Bundle
21 The Splitting Principle and Flag Manifolds

10 Presheaves and Čech Cohomology 11 Sphere Bundles
12 Thom Isomorphism and Poincaré Duality Revisited
13 Monodromy
Spectral Sequences and Applications
14 The Spectral Sequence of a Filtered Complex
22 Pontrjagin Classes
23 The Search for the Universal Bundle
References
List of Notations
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