General Cohomology Theory and K-Theory
These notes constitute a faithful record of a short course of lectures given in São Paulo, Brazil, in the summer of 1968. The audience was assumed to be familiar with the basic material of homology and homotopy theory, and the object of the course was to explain the methodology of general cohomology theory and to give applications of K-theory to familiar problems such as that of the existence of real division algebras. The audience was not assumed to be sophisticated in homological algebra, so one chapter is devoted to an elementary exposition of exact couples and spectral sequences.
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General Cohomology Theories
Chapter HI The Generalized AtiyahHirzebruch Spectral
KTheory the Chern Character and the Hopf
Appendix On the Construction of Cohomology Theories
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abelian groups algebra associated base-point bi-degree bundle g cell-complex ch(z Chapter Chern character Chern classes coefficient group cohomology theory commutative diagram complex vector bundle Consider CW-complex defined denote derived couple dimension axiom direct limit direct sum epimorphism equivalence classes exact couple EC(X exact sequence exactness axiom exists fibration Fibre Bundles filtration finite finite-dimensional follows Fphq(X functor given graded vector space groups and homomorphisms Hence HJ(X hn(X homotopy classes homotopy equivalence Hopf invariant Hp(X hq(X hq(Xp Husemoller induced inverse limit isomorphism K-theory ker(hq(X l-connected Lemma map f mapping cone monic monomorphism morphism n-bundle natural transformation obtained ordinary cohomology pair pretheory Proof prove pullback pushout quotient r+s=q reduced cohomology satisfies semi-group homomorphism shows space over Q spectral sequence spectrum splitting principle subgroup suspension theory h topological trivial bundle unique verify Whitney sum