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Transactions of the American Mathematical Society

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


Authors: V. Giambalvo, David J. Pengelley and Douglas C. Ravenel
Journal: Trans. Amer. Math. Soc. 307 (1988), 433-455
MSC: Primary 55R40; Secondary 55R45, 57R90
DOI: https://doi.org/10.1090/S0002-9947-1988-0940211-9
MathSciNet review: 940211
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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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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1988-0940211-9
Keywords: Classifying space for vector bundles, connected covers of $ BO$, connective cobordism Thom spectra, $ \bmod 2$ homology algebra
Article copyright: © Copyright 1988 American Mathematical Society

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