Zen-ichi Yosimura

A note on the relation of

Let MC/*( ) and K*( ) denote the Z2-graded complex cobordism theory and the complex K-theory respectively. The Thorn homomorphism μc*\ π0(MU)-^ πQ(K) on coefficient groups is identified (up to sign) with the classical Todd genus Td\ Λ-^Z. We denote by / the ideal of Λ to be the kernel of Td: Λ->Z. Wolff [7] proved that the decreasing filtration {IMU*( )} of MU*( ) consists of cohomology theories defined on the category of based finite CPt^-complexes, and the associated quotients IMU*( )[IMU*( ) are determined by the complex ^-theories KG

Z\sb2

( ) with coefficients Gq=I II. The purpose of this note is to extend the Wolffs result to the category of based CW-complexes. Let FqMU be the CPF-spectrum associated with the cohomology theory IMU*( ), i.e., {Y, FqMU}*^I MU*(Y) for any based finite CW-complex (or finite CPF-spectrum). We show that {FgMU*( )} is a decreasing filtration of MC7*( ) consisting of Λ-modules so that the associated quotients are equal to KGf ( ), and in addition that F<I+1MU*( ) is a direct summand of FqMU*( ). Moreover we give a tower