A Fractal-Like Algebraic Splitting of the Classifying Space for Vector Bundles

The connected covers of the classifying space BO induce a decreasing filtration {B,) of H, ( B O ;212) by sub-Hopf algebras over the Steenrod algebra A. We describe a multiplicative grading on H*( B O ;212) inducing a direct sum splitting of B, over A,, where {A,) is the usual (increasing) filtration of A. The pieces in the splittings are finite, and the grading extends that of H,R2S3 which splits it into Brown-Gitler modules. We also apply the grading to the Thomifications {M,) of {B,), where it induces splittings of the corresponding cobordism modules over the entire Steenrod algebra. These generalize algebraically the previously known topological splittings of the connective cobordism spectra M O , M S O and MSpin. Introduction. The classifying space for vector bundles, BO, is of longstanding interest in topology. We will describe a splitting of the mod 2 homology algebra of BO, having applications to connective cobordism Thom spectra. The splitting will be multiplicative; in other words it will be fully compatible with Whitney sums of vector bundles. It differs from other familiar splittings in topology in the way it interacts with the connected covers of BO and the Steenrod algebra A of cohomology operations. We will explain how this interaction is analogous to the geometric properties of the boundary of the fractal Mandelbrot set (or M-set) [PR]. The boundary of the M-set has two attributes: First, patterns become more elaborate upon magnification. Second, patterns visible at one level of magnification actually reappear under further magnification (self-similarity). The second property is Mandelbrot's idea of a fractal structure [MI,while the first is an additional feature of certain fractals, like the boundary of the M-set. Our results about certain subalgebras of the algebra H, BO over the Hopf algebra A reveal precisely these two features. Specifically, consider the decreasing algebra filtration {B,) of H, BO provided by the images of the connected covers, and the standard increasing Hopf algebra filtration {A,) of A, where A, is generated by the first 2, Steenrod squares. The analogy to the geometric properties of the boundary of the M-set is now made precise by interpreting "pattern" to mean a multiplicative direct sum splitting of an algebra B, over the Hopf algebra A,, "magnification" as descending in the filtration {B,), and "more elaborate" as ascending in the filtration {A,). Received by the editors May 15, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 55R40, 55R45, 571390, 57T05.