zbMATH — the first resource for mathematics

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

55P42 Stable homotopy theory, spectra
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology
20G05 Representation theory for linear algebraic groups
Full Text: DOI
[1] May, Amer. Math. Soc. Cont. Math. 37 pp 121– (1985)
[2] DOI: 10.2307/2374436 · Zbl 0551.55005 · doi:10.2307/2374436
[3] Curtis, Methods of Representation Theory I (1981)
[4] Clifford, Algebraic Theory of Semigroups 1 (1961) · Zbl 0111.03403 · doi:10.1090/surv/007.1
[5] DOI: 10.2307/2006940 · Zbl 0586.55008 · doi:10.2307/2006940
[6] Carlisle, J. Algebra
[7] DOI: 10.1016/0040-9383(85)90014-X · Zbl 0611.55010 · doi:10.1016/0040-9383(85)90014-X
[8] Toda, Mem. Coll. See. Kyoto 32 pp 297– (1959)
[9] Adams, Infinite Loop Spaces 90 (1978) · Zbl 0398.55008 · doi:10.1515/9781400821259
[10] Swan, Illinois J. Math. 4 pp 341– (1960)
[11] Lallemont, Trans Amer. Math. Soc. 139 pp 392– (1969)
[12] Kuhn, Algebraic Topology ? Proceedings, Seattle pp 286– (1985)
[13] DOI: 10.2307/2000710 · Zbl 0637.55002 · doi:10.2307/2000710
[14] DOI: 10.2307/2000005 · Zbl 0544.55016 · doi:10.2307/2000005
[15] James, The Representation Theory of the Symmetric Group 16 (1981)
[16] Huppert, Endliche Gruppen I (1967) · Zbl 0217.07201 · doi:10.1007/978-3-642-64981-3
[17] DOI: 10.1016/0021-8693(81)90309-4 · Zbl 0474.20021 · doi:10.1016/0021-8693(81)90309-4
[18] Howie, An Introduction to Semigroup Theory (1976) · Zbl 0355.20056
[19] DOI: 10.1016/0021-8693(78)90116-3 · Zbl 0376.20008 · doi:10.1016/0021-8693(78)90116-3
[20] Serre, Linear Representation of Finite Groups 42 (1977) · doi:10.1007/978-1-4684-9458-7
[21] Rotman, The Theory of Groups (1973) · Zbl 0262.20001
[22] DOI: 10.1007/BF01246939 · Zbl 0547.55017 · doi:10.1007/BF01246939
[23] DOI: 10.1016/0040-9383(83)90014-9 · Zbl 0526.55010 · doi:10.1016/0040-9383(83)90014-9
[24] Landrock, Finite Group Algebras and their Modules (1983) · doi:10.1017/CBO9781107325524
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.