Differential Forms in Algebraic TopologySpringer 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. |
Índice
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 | |
Otras ediciones - Ver todo
Differential Forms in Algebraic Topology Raoul Bott,Loring W. Tu No hay ninguna vista previa disponible - 2011 |
Differential Forms in Algebraic Topology Raoul Bott,Loring W. Tu No hay ninguna vista previa disponible - 2014 |
Differential Forms in Algebraic Topology Raoul Bott,Loring W. Tu No hay ninguna vista previa disponible - 1995 |
Términos y frases comunes
Abelian group Čech cohomology Chern classes cochain cocycle coefficients cohomology class compact support complex vector bundle constant presheaf coordinate ℂP defined definition diffeomorphic dimension direct sum double complex element EMARK Euler class exact sequence example Exercise fiber bundle fibration Figure finite good cover follows functor given global form Grassmannian Hence homology homomorphism homotopy groups inclusion induces integral Künneth formula Lemma line bundle locally constant map f MayerVietoris sequence neighborhood ofthe open cover open sets oriented manifold oriented vector bundle partition of unity Poincaré dual Poincaré duality presheaf projective space proof Proposition pullback rank Rham cohomology Rham complex Rham theory ℝn ROOF singular singular cohomology spectral sequence sphere bundle surjective tangent bundle theorem Thom class Thom isomorphism topological space transition functions trivial vector space zero section αβ ϕ α