Homology Operations on a New Infinite Loop Space

Boyer et al. [1] defined a new infinite loop space structure on the space Mq = fTn>i ^(Z> 2/i) such that the total Chern class map BU —► Mq is an infinite loop map. This is a sort of Riemann-Roch theorem without denominators: for example, it implies Fulton-MacPherson's theorem that the Chem classes of the direct image of a vector bundle E under a given finite covering map are determined by the rank and Chern classes of E. We compute the Dyer-Lashof operations on the homology of Mq ■ They provide a new explanation for Kochman's calculation of the operations on the homology of BU , and they suggest a possible characterization of the infinite loop structure on Mq ■ The total Chern class of the direct sum of two vector bundles, c(E © F), is not the sum but rather the product of the total Chern classes: c(E © F) = c(E)c(F). In other words, for a topological space X, the total Chern class map c: K°X —> Iln>i H2n(X; Z) is not a homomorphism of abelian groups for the obvious additive group structure on the set Un>x H2n(X; Z), but only for a multiplicative abelian group structure on it; let M°X = 1 + n„>i Hln(X; Z) with the appropriate multiplicative group structure. Since the group K°X is the Oth term of a cohomology theory k*X (O-connective AT-theory), it is natural to expect that M°X is also the Oth term of a^cohomology theory M*X, with the property that the map_c: k°X = K°X -» M°X extends to a map of cohomology theories k*X -» M*X. Recently Boyer, Lawson, Lima-Filho, Mann, and Michelsohn [1] defined a cohomology theory M* with this property. If Mo denotes the infinite loop space such that [X, Mo] = M°X, then Mq ~ IIn>i -^(Z, 2n) as a space, and we know the //"-space structure on Mq . (It is not the usual //-space structure on a product of Eilenberg-Mac Lane spaces.) In this paper we continue the analysis of the infinite loop space Mo by computing the Araki-Kudo-Dyer-Lashof operations on Ht(Mo; Z/p). The computation starts with Kochman's calculation of these operations on H*BU [4], but there is a pleasant side effect to our analysis. We only need Kochman's calculation in the lowest nontrivial dimension, and with that as input, the rich structure of Mo (it is the multiplicative Eoo space which underlies one component of an E^ ring space) allows us to find all the operations on HtMo and hence on H*BU. In particular, one could say that the existence of BLLMM's theory gives a conceptual explanation for the simplicity of Kochman's formulas. Received by the editors August 20, 1993. 1991 Mathematics Subject Classification. Primary 55S12; Secondary 55P47. This paper was supported by an NSF postdoctoral fellowship. ©1995 American Mathematical Society 99 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use