Raoul Bott Loring W. Tu Differential Forms in Algebraic Topology

With 92 Illustrations

Springer-Verlag New York Heidelberg Berlin Contents

Introduction 1

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 Q* 19 The Mayer-Vietoris sequence 22 The functor fif 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 Poincare Lemmas 33 The Poincare lemma for de Rham cohomology 33 The Poincare 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 Poincare duality on an orientable manifold 44

XI xii Contents

The Kiinneth formula and the Leray-Hirsch theorem 47 The Poincare dual of a closed oriented submanifold 50 §6 The Thorn 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 Poincare duality and the Thorn 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, Poincare duality, and the Thom isomorphism 85

CHAPTER II The Cech-de Rham Complex 89

§8 The Generalized Mayer-Vietoris Principle 89 Reformulation of the Mayer-Vietoris 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 Cech 102 The tic-tac-toe proof of the Kiinneth formula 105

§10 Presheaves and Cech Cohomology 108 Presheaves 108 Cech cohomology 110

§11 Sphere Bundles Orientability The Euler class of an oriented sphere bundle The global angular form Euler number and the isolated singularities of a section Euler characteristic and the Hopf index theorem

§12 The Thom Isomorphism and Poincare Duality Revisited The Thom isomorphism Euler class and the zero locus of a section A tic-tac-toe lemma Poincare duality Contents Xlll

§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 2 7t3(S ) 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 3 Computation of 7t4(S ) 251 xiv Contents

The Whitehead tower 252 3 Computation of 7is(S ) 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 Chern 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