| introduction . chapter i de rham theory 1 the de rham complex on rn the de rham complex compact supports 2 the mayer-vietoris sequence the functor ω* the mayer-vietoris sequence the functor ω* and the mayer-vietoris sequence for compact supports 3 orientation and integration orientation and the integral of a differential form stokes' theorem 4 poincare lemmas the poincare lemma for de rham cohomology the poincare lemma for compactly supported cohomoiogy the degree of a proper map 5 the mayer-vietoris argument existence of a good cover finite dimensionality of de rham cohomoiogy .poincare duality on an orientable manifold the kunneth formula and the leray-hirsch theorem the poincare dual of a closed oriented submanifold 6 the thom isomorphism vector bundles and the reduction of structure groups operations on vector bundles compact cohomology of a vector bundle compact vertical cohomology and integration along the fiber poincare duality and the thom class the global angular form, the euler class, and the thom class relative de rham theory 7 the nonorientable case the twisted de rham complex integration of densities, poincare duality, and the thom isomorphism chapter ii the cech-de rham complex 8 the generalized mayer-vietoris principle reformulation of the mayer-vietoris sequence generalization to countably many open sets and applications 9 more examples and applications of the mayer-vietoris principle examples: computing the de rham cohomology from the combinatorics of a good cover explicit isomorphisms between the double complex and de rham and cech the tic-tac-toe proof of the kunneth formula 10 presheaves and cech cohomology presheaves cech cohomology 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 poincar6 duality 13 monodromy when is a locally constant presheaf constant?.. examples of monodromy chapter iii spectral sequences and applications 14 the spectral sequence of a filtered complex exact couples the spectral sequence of a filtered complex the spectral sequence of a double complex the spectral sequence of a fiber bundle some applications product structures the gysin sequence leray's construction 15 cohomology with integer coefficients singular homology the cone construction the mayer-vietoris sequence for singular chains singular cohomology the homology spectral sequence 16 the path fibration the path fibration the cohomology of the loop space of a sphere 17 review of homotopy theory homotopy groups the relative homotopy sequence some homotopy groups of the spheres attaching cells digression on morse theory the relation between homotopy and homology π3(s2) and the hopf invariant 18 applications to homotopy theory eilenberg-maclane spaces the telescoping construction the cohomology of k(z, 3) the transgression basic tricks of the trade postnikov approximation computation of π4(s3) the whitehead tower computation of π5(s3) 19 rational homotopy theory minimal models examples of minimal models the main theorem and applications chapter iv characteristic classes 20 chern classes of a complex vector bundle the first chern class of a complex line bundle the projectivization of a vector bundle main properties of the chern classes 21 the splitting principle and flag manifolds the splitting principle proof of the whitney product formula and the equality of the top chern class and the euler class computation of some chern classes flag manifolds 22 pontrjagin classes conjugate bundles realization and complexification the pontrjagin classes of a real vector bundle application to the embedding of a manifold in a euclidean space 23 the search for the universal bundle the grassmannian digression on the poincare series of a graded algebra the classification of vector bundles the infinite grassmannian concluding remarks references list of notations index ... |
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