A fractal-like algebraic splitting of the classifying space for vector bundles

Giambalvo, V.; Pengelley, David J.; Ravenel, Douglas C.

doi:10.1090/S0002-9947-1988-0940211-9

A fractal-like algebraic splitting of the classifying space for vector bundles
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by V. Giambalvo, David J. Pengelley and Douglas C. Ravenel PDF

Trans. Amer. Math. Soc. 307 (1988), 433-455 Request permission

Abstract:

The connected covers of the classifying space $BO$ induce a decreasing filtration $\{ {B_n}\}$ of ${H_{\ast }}(BO; Z/2)$ by sub-Hopf algebras over the Steenrod algebra $A$. We describe a multiplicative grading on ${H_{\ast }}(BO; Z/2)$ inducing a direct sum splitting of ${B_n}$ over ${A_n}$, where $\{ {A_n}\}$ is the usual (increasing) filtration of $A$. The pieces in the splittings are finite, and the grading extends that of ${H_{\ast }}{\Omega ^2}{S^3}$ which splits it into Brown-Gitler modules. We also apply the grading to the Thomifications $\{ {M_n}\}$ of $\{ {B_n}\}$, 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 $MO$, $MSO$ and $M Spin$.

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