A fractal-like algebraic splitting of the classifying space for vector bundles. (English) Zbl 0666.55012
All homology is with Z/2 coefficients. Recall that \(A_ n\) is the subalgebra of the mod 2 Steenrod algebra, A, generated by the first \(2^ n\) squares. \(A_ 0\subset A_ 1\subset A_ 2\subset...\subset A\) forms an ascending filtration of A. Let \(B0<\phi (n)>\) be the n-th distinct connective cover of BO. Let \(B_ n\) be the image of \(H_*BO<\phi (n)>\to H_*BO\). \(B_ 0=H_*BO\supset B_ 1=H_*BSO\supset B_ 2=H_*BSpin\supset B_ 3=H_*BO<8>..\). forms a descending filtration of \(H_*BO\). The authors give new polynomial generators \(u_ i\) for \(H_*BO=Z/2[u_ 1,u_ 2,...]\) which enjoy nice properties with respect to the \(A_ n\) actions on the \(\{B_ n\}\) filtration. Specifically: \(B_ 1=Z/2[u^ 2_ 1,u^ 2_ 2,u_ 3,u^ 2_ 4,u_ 5...]\); \(B_ 2=Z/2[u^ 4_ 1,u^ 4_ 2,u^ 2_ 3,u^ 4_ 4,u^ 2_ 5,u^ 2_ 6,u_ 7,u^ 4_ 8,...]\); \(B_ 3=Z/2[u^ 8_ 1,u^ 8_ 2,u^ 4_ 3,...]\). (The exponents of the \(u_ i\) depend on i’s dyadic expansion and on the size of i with respect to n.) An important subalgebra of \(H_*BO\) which slices across the \(B_ n's\) is \(H_*\Omega^ 2S^ 3=Z/2[u_ 1,u_ 3,u_ 7,...]\). F. Cohen, Th. J. Lada and J. P. May: The homology of iterated loop spaces [Lect. Notes Math. 533 (1976; Zbl 0334.55009)] gave an algebra grading on \(H_*\Omega^ 2S^ 3\) yielding an A-module splitting of \(H_*\Omega^ 2S^ 3\) into what are now recognized as Brown-Gitler modules. The authors’ new generators allow them to extend Cohen’s grading (doubled) to an algebra grading of \(H_*BO\). When this extended grading is restricted to \(B_ n\), it is \(A_ n\)-invariant; this leads to an \(A_ n\)-splitting of \(B_ n\). Hence the smaller pieces of \(H_*BO\) split over larger pieces of A. As its title indicates, this work is encapsulated in the ideas of fractal technology.
Reviewer: D.C.Johnson
MSC:
| 55R40 | Homology of classifying spaces and characteristic classes in algebraic topology |
| 55R45 | Homology and homotopy of \(B\mathrm{O}\) and \(B\mathrm{U}\); Bott periodicity |
| 57R90 | Other types of cobordism |
| 57T05 | Hopf algebras (aspects of homology and homotopy of topological groups) |
| 55N15 | Topological \(K\)-theory |
| 55P42 | Stable homotopy theory, spectra |
Keywords:
classifying space for real K-theory; polynomial generators for \(H_ *BO\); mod 2 Steenrod algebra; n-th distinct connective cover of BO; Brown-Gitler modules; algebra grading of \(H_ *BO\); \(A_ n\)-splitting of \(B_ n\)Citations:
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