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With ~'i the Hopf Sp (1) -bundle over S4. ~4 t (X). Then(@~(S4k+2 ) = s x t4k SU yields the information necessary to prGve the remark. ) U = Kn(x). > ~U(-), and commutatlvlty 30 6. The~ ~ ~c" After discussing the cohomology of MU(n) and the classical Thom isomorphism theorem, we go on to associate with each element of H~-~(BU) a homombrphlsm~*u(X) > H~(X) . In terms of these homomorphisms we can characterize the composite fl*tX) U ~c > K*(X) where ch is Chern character. ~ H*(X;Z) In particular, for X a point, the composite I~* (pt) ~ K° (pt) = Z U is characterized in terms of the classical Todd genus [16] and t h u s / ~ c is determined on the coefficient groups.

We may as well suppose m even. The Todd genus Td [M2n] of Hirzebruch is the similar number using Q(tl'''"tm) tI • • •tm (1 - exp (-tl))'"(l - exp (-tm)) Note that P(tl,--°,tm) = Q(-tl,---,-tm). The corollary follows readily. 38 CHAPTER II. COBORDISM CHARACTERISTIC CLASSES. ) be a given multiplicatlve cohomology theory. The main purpose of section 7 is to give the general sufficient conditions so that we may be able to assign to every Sp(m)-bundle ~ over a finite CW complex X, an element p(~) in h* (X) where pk ( ~ ) ¢ h4 = i + pl Q~) + ' ' ' + k(x ) .

There is also ch : K (X) --~ H (,Q) X" mapping K2k(x) into HeV(X;Q) and K2k+l(x) into H°d(x;Q). ring homomorphism given by the composite The natural 53 is denoted by ph : Ko*(x) --~ B*(x;Q). It follows by induction on k that K0*(S k) is a free K0*-module with a basis consisting of one element ~ of K0k(sk), namely any element with ph ~ the image of a generator under Hk(sk;z) ~> Hk(sk;Q). We also need a little information concerning KSp(X) = KSp°(X). There is the product > KO(X) KSp(X) ~ KSp(X) mapping ( ~ , 7 ) into the tensor product ~ ~H ~ as in section 3.

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