Differential forms in algebraic topology:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | German |
| Published: |
New York [u.a.]
Springer
1995
|
| Edition: | Corr. 3. printing |
| Series: | Graduate texts in mathematics
82 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke, Literaturverz. S. 307 - 310 |
| Physical Description: | XIV, 331 S. graph. Darst. |
| ISBN: | 9780387906133 0387906134 |
Staff View
MARC
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| 100 | 1 | |a Bott, Raoul |d 1923-2005 |e Verfasser |0 (DE-588)119134136 |4 aut | |
| 245 | 1 | 0 | |a Differential forms in algebraic topology |c Raoul Bott ; Loring W. Tu |
| 250 | |a Corr. 3. printing | ||
| 264 | 1 | |a New York [u.a.] |b Springer |c 1995 | |
| 300 | |a XIV, 331 S. |b graph. Darst. | ||
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| 490 | 1 | |a Graduate texts in mathematics |v 82 | |
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke, Literaturverz. S. 307 - 310 | ||
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| 650 | 4 | |a Differential forms | |
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Record in the Search Index
| _version_ | 1804124813359316992 |
|---|---|
| adam_text | Contents
Introduction
ι
CHAPTER I
De Rham
Theory
13
§1
The
de Rham
Complex on R
13
The
de Rham
complex
13
Compact supports
17
§2
The Mayer-Vietoris Sequence
19
The functor
Ω*
19
The Mayer-Vietoris sequence
22
The functor
Ω*
and the Mayer-Vietoris sequence for compact supports
25
§3
Orientation and Integration
27
Orientation and the integral of a differential form
27
Stokes theorem
31
§4
Poincaré
Lemmas
33
The
Poincaré
lemma for
de Rham
cohomology
33
The
Poincaré
lemma for compactly supported cohomology
37
The degree of a proper map
40
§5
The Mayer-Vietoris Argument
42
Existence of a good cover
42
Finite dimensionality of
de Rham
cohomology
43
Poincaré
duality on an
orientable
manifold
44
Xl
XU Contents
The Künneth
formula and the Leray-Hirsch theorem
47
The
Poincaré
dual of a closed oriented submanifold
50
§6
The Thom Isomorphism
53
Vector bundles and the reduction of structure groups
53
Operations on vector bundles
56
Compact cohomology of a vector bundle
59
Compact vertical cohomology and integration along the fiber
61
Poincaré
duality and the Thom class
65
The global angular form, the
Euler
class, and the Thom class
70
Relative
de Rham
theory
78
§7
The Nonorientable Case
79
The twisted
de Rham
complex
79
Integration of densities,
Poincaré
duality, and the Thom isomorphism
85
CHAPTER II
The
Čech-de Rham
Complex
89
§8
The Generalized Mayer-Vietoris Principle
89
Reformulation of the Mayer-Vietoiis sequence
89
Generalization to countably many open sets and applications
92
§9
More Examples and Applications of the Mayer-Vietoris Principle
99
Examples: computing the
de Rham
cohomology from the
combinatorics of a good cover
100
Explicit isomorphisms between the double complex and
de Rham
and
Čech
102
The tic-tac-toe proof of the
Künneth
formula
105
§10
Presheaves and
Čech
Cohomology
108
Presheaves
108
Čech
cohomology
110
§11
Sphere Bundles
113
Orientability
114
The
Euler
class of an oriented sphere bundle
116
The global angular form
121
Euler
number and the isolated singularities of a section
122
Euler
characteristic and the
Hopf
index theorem
126
§12
The Thom Isomorphism and
Poincaré
Duality Revisited
129
The Thom isomorphism
130
Euler dass
and the zero locus of a section
133
A tic-tac-toe lemma
135
Poincaré
duality
139
Contents xiii
§13
Monodromy
141
When is a locally constant presheaf constant?
141
Examples of monodromy
151
CHAPTER III
Spectral Sequences and Applications
154
§14
The Spectral Sequence of a Filtered Complex
155
Exact couples
155
The spectral sequence of a filtered complex
156
The spectral sequence of a double complex
161
The spectral sequence of a fiber bundle
169
Some applications
170
Product structures
174
The Gysin sequence
177
Leray s construction
179
§15
Cohomology with Integer Coefficients
182
Singular homology
183
The cone construction
184
The Mayer-Vietoris sequence for singular chains
185
Singular cohomology
188
The homology spectral sequence
196
§16
The Path Fibration
197
The path fibration
198
The cohomology of the loop space of a sphere
203
§17
Review of Homotopy Theory
206
Homotopy groups
206
The relative homotopy sequence
212
Some homotopy groups of the spheres
213
Attaching cells
217
Digression on Morse theory
220
The relation between homotopy and homology
225
п3{Ѕг)
and the
Hopf
invariant
227
§18
Applications to Homotopy Theory
239
Eilenberg-MacLane spaces
240
The telescoping construction
241
The cohomology of K(Z,
3) 245
The transgression
247
Basic tricks of the trade
249
Postnikov approximation
250
Computation of
л4(Ѕ3)
251
xiv Contents
The Whitehead tower
252
Computation of n,{S3)
256
§19
Rational Homotopy Theory
258
Minimal models
259
Examples of Minimal Models
259
The main theorem and applications
262
CHAPTER IV
Characteristic Classes
266
§20
Chem
Classes of a Complex Vector Bundle
267
The first Chern class of a complex line bundle
267
The projectivization of a vector bundle
269
Main properties of the Chern classes
271
§21
The Splitting Principle and Flag Manifolds
273
The splitting principle
273
Proof of the Whitney product formula and the equality
of the top Chern class and the
Euler
class
275
Computation of some Chern classes
278
Flag manifolds
282
§22
Pontrjagin Classes
285
Conjugate bundles
286
Realization and complexification
286
The Pontrjagin classes of a real vector bundle
289
Application to the embedding of a manifold in a
Euclidean space
290
§23
The Search for the Universal Bundle
291
The Grassmannian
292
Digression on the Poincare series of a graded algebra
294
The classification of vector bundles
297
The infinite Grassmannian
302
Concluding remarks
303
References
307
List of Notations
311
Index
319
|
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| author | Bott, Raoul 1923-2005 Tu, Loring W. 1952- |
| author_GND | (DE-588)119134136 (DE-588)110090322 |
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| discipline | Mathematik |
| edition | Corr. 3. printing |
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| id | DE-604.BV010387544 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T17:51:38Z |
| institution | BVB |
| isbn | 9780387906133 0387906134 |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006916500 |
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| publisher | Springer |
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| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Bott, Raoul 1923-2005 Verfasser (DE-588)119134136 aut Differential forms in algebraic topology Raoul Bott ; Loring W. Tu Corr. 3. printing New York [u.a.] Springer 1995 XIV, 331 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 82 Hier auch später erschienene, unveränderte Nachdrucke, Literaturverz. S. 307 - 310 Algebraic topology Differential forms Differential topology Algebraische Topologie (DE-588)4120861-4 gnd rswk-swf Differentialtopologie (DE-588)4012255-4 gnd rswk-swf Differentialform (DE-588)4149772-7 gnd rswk-swf Differentialform (DE-588)4149772-7 s Algebraische Topologie (DE-588)4120861-4 s DE-604 Differentialtopologie (DE-588)4012255-4 s 1\p DE-604 Tu, Loring W. 1952- Verfasser (DE-588)110090322 aut Graduate texts in mathematics 82 (DE-604)BV000000067 82 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006916500&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Bott, Raoul 1923-2005 Tu, Loring W. 1952- Differential forms in algebraic topology Graduate texts in mathematics Algebraic topology Differential forms Differential topology Algebraische Topologie (DE-588)4120861-4 gnd Differentialtopologie (DE-588)4012255-4 gnd Differentialform (DE-588)4149772-7 gnd |
| subject_GND | (DE-588)4120861-4 (DE-588)4012255-4 (DE-588)4149772-7 |
| title | Differential forms in algebraic topology |
| title_auth | Differential forms in algebraic topology |
| title_exact_search | Differential forms in algebraic topology |
| title_full | Differential forms in algebraic topology Raoul Bott ; Loring W. Tu |
| title_fullStr | Differential forms in algebraic topology Raoul Bott ; Loring W. Tu |
| title_full_unstemmed | Differential forms in algebraic topology Raoul Bott ; Loring W. Tu |
| title_short | Differential forms in algebraic topology |
| title_sort | differential forms in algebraic topology |
| topic | Algebraic topology Differential forms Differential topology Algebraische Topologie (DE-588)4120861-4 gnd Differentialtopologie (DE-588)4012255-4 gnd Differentialform (DE-588)4149772-7 gnd |
| topic_facet | Algebraic topology Differential forms Differential topology Algebraische Topologie Differentialtopologie Differentialform |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006916500&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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