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Stable decompositions of classifying spaces of finite Abelian p-groups. (English) Zbl 0686.55007
In this paper, the authors consider the problem of finding a stable decomposition BG\(\simeq X_ 1\vee...\vee X_ N\) into indecomposable wedge summands, where BG is the classifying space of a finite group G. It suffices to consider this splitting problem one prime at a time (i.e., where G is a p-group P), and the authors study the case where P is abelian. Questions about this case are then transformed into purely group theoretic ones about the finite general linear groups. This splitting problem is then one of idempotent decompositions of 1 in \(\{BP_+,BP_+\}\) or (since BP is p-adically complete) in the ring \(\{BP_+,BP_+\}\otimes {\mathbb{F}}_ p\), which has been calculated as a finite-dimensional algebra, as a consequence of the Segal conjecture [J. P. May, Conf. Algebraic Topology 1983, Contemp. Math. 37, 121- 129 (1985; Zbl 0565.55021)]. By reducing this ring to simpler rings, and from general representation theory, the authors prove the following result [Theorem A(1)]: In a complete stable decomposition of \(B({\mathbb{Z}}/p)^ n_+\) there are wedge summands of \(p^ n\) distinct homotopy types. These correspond to the irreducible right \({\mathbb{F}}_ p[M_ n({\mathbb{Z}}/p)]\)-modules, and a given homotopy type appears with multiplicity equal to the dimension of the corresponding module. The splitting induces a direct sum decomposition of \(H*(({\mathbb{Z}}/p)^ n)\) into indecomposable modules over the Steenrod algebra A.
Similarly, the authors consider the classification of classifying spaces BG, up to stable equivalence, where G has a given abelian p-Sylow subgroup P. They introduce a certain Grothendieck group and determine its structure (Theorem B). Finally, they study a particularly interesting wedge summand of \(B({\mathbb{Z}}/p)^ n\) and describe some of its properties (Theorem C), and they give a number of interesting examples in the final section of the paper.
Reviewer: S.Thomeier

MSC:
55P42 Stable homotopy theory, spectra
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology
20G05 Representation theory for linear algebraic groups
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