The Hopf Ring for Complex Cobordism

It is our purpose here to announce the results of our study of the homology of the spaces in the £2-spectrum for complex cobordism and BrownPeterson cohomology. Let MU(n) be the standard Thorn complex. MUk = l im^^ ü,(~~)MU(n) is the 2k space in the £2-spectrum for complex cobordism. We will consider the space MU — hmn^_00 Iï ;>n MUf-. We find this product easier to study than the separate factors, as will become apparent below. For a space X we have [X, MU] ~ U*(X), the even degree part of the complex cobordism of X. Because MU is a multiplicative theory, U *(X) is a ring and MU is a commutative ring with identity in the homotopy category. Thus we have that for any field kt H%(MU; k) is a commutative ring with identity in the category of fc-coalgebras, i.e., it is a "Hopf ring". In more common language, the homology has two products and a coproduct. o will denote the multiplicative product which comes from the ring structure on the spectrum, while * will denote the additive product coming from the loop structure (Ü,MU — MU). They obey the following distributive law: if \p(z) = 2 z ® z" is the coproduct, then z ° (x * j ) =