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Global structure of the mod two symmetric algebra, \(H^\ast(BO;\mathbb F_2)\) over the Steenrod Algebra. (English) Zbl 1057.55004

The authors determine a mimimal presentation of the mod \(2\) cohomology of the classifying space \(BO\) within the category of unstable algebras over the mod \(2\) Steenrod algebra \({\mathcal A}\). More specifically they show that \(H^*(BO)\) is isomorphic to the free unstable \({\mathcal A}\)-algebra generated by the Stiefel-Whitney classes \(\{ w_{2^k}; k\geq 0\}\) modulo the ideal generated by a two-parameter family of relations \(\{\theta_{i,j}; i,j \geq 0\}\), where \(\theta_{i,j}\) expresses the sum \(Sq^{2^j} w_{2^i} + Sq^{2^{i-1}} Sq^{2^j} w_{2^{i-1}}\) as a sum of decomposables.
The authors then use this result to find corresponding minimal unstable \({\mathcal A}\)-algebra presentations also for \(H^*(BO(m)), m\geq 1\) and for the subalgebras \(B^*(n) \subset H^*(BO\langle \phi(n) \rangle), n\geq 1\), where \(B^*(n)\) by definition is the image of the map in mod \(2\) cohomology induced by the canonical map from the \(n\)-th distinct connected cover \(BO\langle \phi(n) \rangle\) of \(BO\) down to \(BO\).
As an application of the latter they show that the pushout of the homomorphisms \(H^*(BO) \to H^*(BO(2^{n+1}-1))\) and \(H^*(BO) \to B^*(n)\) can be identified with the \(n\)-th Dickson algebra \(W_{n+1}\), thereby using a characterization of the Dickson algebra \(W_{n+1}\) given in [D. Pengelley, F. Peterson and F. Williams, Math. Proc. Camb. Philos. Soc. 129, 263–275 (2000; Zbl 0970.55010)]. In addition they use their results to provide a mimimal unstable \({\mathcal A}\)-module presentation for \(H^*({\mathbb R}P^\infty)\) as well as for some of its subquotients. The corresponding proofs are based on calculations involving the Kudo-Araki-May algebra, which already has been investigated in detail by the authors in [Trans. Am. Math. Soc. 352, 1453–1492, (2000; Zbl 0946.55012)].

MSC:

55R45 Homology and homotopy of \(B\mathrm{O}\) and \(B\mathrm{U}\); Bott periodicity
13A50 Actions of groups on commutative rings; invariant theory
16W22 Actions of groups and semigroups; invariant theory (associative rings and algebras)
16W50 Graded rings and modules (associative rings and algebras)
55R40 Homology of classifying spaces and characteristic classes in algebraic topology
55S05 Primary cohomology operations in algebraic topology
55S10 Steenrod algebra
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