Homology operations on a new infinite loop space

Abstract

Boyer et al. [1] defined a new infinite loop space structure on the space M 0 = ∏ n ⩾ 1 K ( Z , 2 n ) {M_0} = {\prod _{n \geqslant 1}}K({\mathbf {Z}},2n) such that the total Chern class map B U → M 0 BU \to {M_0} 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 Chern classes of the direct image of a vector bundle E E under a given finite covering map are determined by the rank and Chern classes of E E . We compute the Dyer-Lashof operations on the homology of M 0 {M_0} . They provide a new explanation for Kochman’s calculation of the operations on the homology of B U BU , and they suggest a possible characterization of the infinite loop structure on M 0 {M_0} .