New PDF release: Topics in K-Theory

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A. , 7. G. Bredon, 8. H. Cartan, 9. i0. A. Dold, , and S. ii. Eilenberg, and R. Lashof, 12. 13. E. Dyer, L. Hodgkin, 14. M. James, 15. S. MacLane, 15a. 16. J. Milnor, 17. The K-theory of Eilenberg-MacLane complexes, To o _ ~ 7 (1968), 317-329. Characters and cohomology of finite groups, Publ. Math. I H E S 9 (1961), 23-64. Vector bundles and the Kunneth formula, ToDolo~ i (1962), 245-248. K-theory, Benjamin, 1967. On H-spaces and infinite loop spaces. Category Theory, Homology Theory and their Applications III, Lecture Note~ in Mathematics no.

Ii) If in addition X or Y is a free G-space, then $ is itself an isomorphism (iii) If X or Y is a trivial G-s~ace, This result, theorems with those already proved in w claimed in the Introduction. 1, a generalization main work of the section is a reduction of parts case X = Y = G; fo__frany G. of part (iii). , the 'obstruction to con@ F(X,Y;KG) , which we shall write F(X,Y) from now on, is defined; that vergence' it fits into an exact triangle of ~ - 2 -graded R(G)-modules H(X,Y)~ > KG(X • Y) b/ (i) r(x,Y) and is a c o h o m o l o ~ theory in X and Y separately.

N n P2 n It remains, therefore, ~B) is a Kunneth space for h B in Top/B. ) I shall give only a necessary condition, which can probably be weakened; and then d i s c u s s how i n c l u s i v e t h i s c o n d i t i o n i s . 2. _~rhB, * i n Top/B f o r a l l n ~ O. (Dn(X) x B Proof. First, let k: B § Dn(x) be the cross-section; then we can use inductively the isomorphism induced by r n % = h*(Dn+l(x),k(B)) + h*(Dn(X) x B, Cn(Dn(X))) and standard cohomology theory methods to establish that Dn(x) is a Kunneth space for h in Top, not a T o p for all n.

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