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On certain homotopy actions of general linear groups on iterated products. (English) Zbl 0990.55003

Several important advances in homotopy theory have used wedge decompositions or splittings to construct spaces with desirable properties. For example one often seeks spaces whose mod-\(p\) cohomology is free over a subalgebra of the Steenrod algebra. Previous work along these lines has considered iterated products of the abelian \(H\)-spaces \(X=B (Z/p^rZ)\) and \(B(Z_p)\). Here the authors show that the same sort of splitting occurs when \(X\) satisfies much weaker requirements. As applications they give decompositions for iterated products of \(\Omega^2 S^3\), SO(4), and \(G_2\). The latter two are not even homotopy commutative.

MSC:

55P45 \(H\)-spaces and duals
20C20 Modular representations and characters
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology
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