×

Indecomposable A-module summands in H*V which are unstable algebras. (English) Zbl 0686.55010

Let p be a prime and denote by A the mod p Steenrod algebra. We determine the indecomposable A-module summands of \(H*(({\mathbb{Z}}/p)^ d;{\mathbb{F}}_ p)\) which admit the structure of an unstable A-algebra. In fact, it turns out that this is equivalent to the problem of determining those indecomposable A-module summands which arise as the mod p cohomology of a space (or even a classifying space of a finite group). We reduce this problem to one in modular representation theory, namely for which d and p is the projective cover of the trivial one dimensional \(GL(d,{\mathbb{F}}_ p)\) representation \({\mathbb{F}}_ p\) a permutation module? Our solution to this latter problem makes use of the classification of subgroups of \(GL(d,{\mathbb{F}}_ p)\) acting transitively on \(({\mathbb{F}}_ p)^ d\setminus \{0\}\) and hence depends on the classification of finite simple groups (on Feit-Thompson’s odd order theorem if \(p=2)\).
Reviewer: L.Schwartz

MSC:

55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology
55S10 Steenrod algebra
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Adams, J.F., Gunawardena, J.H., Miller, H.R.: The Segal conjecture for elementary abelianp-groups. Topology24, 435–460 (1985) · Zbl 0611.55010 · doi:10.1016/0040-9383(85)90014-X
[2] Bousfield, A.K., Kan, D.M.: Homotopy limits, completions and localizations. (Lect. Notes Math., vol. 304) Berlin Heidelberg New York: Springer 1972 · Zbl 0259.55004
[3] Carlsson, G.: G.B. Segal’s Burnside ring conjecture for (\(\mathbb{Z}\)/2) k . Topology22, 83–103 (1983) · Zbl 0504.55011 · doi:10.1016/0040-9383(83)90046-0
[4] Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of finite groups. Oxford: Clarendon Press 1985 · Zbl 0568.20001
[5] Curtis, C., Reiner, I.: Methods of representation theory I. New York: Wiley 1981 · Zbl 0469.20001
[6] Dickson, L.E.: Linear groups with an exposition of the Galois field theory. Stuttgart: Teubner 1901; reprinted Dover 1958 · JFM 32.0128.01
[7] Foulser, D.A., Kallaher, M.J.: Solvable flag transitive rank 3 collineation groups. Geom. Dedicata7, 111–130 (1978) · Zbl 0406.51009 · doi:10.1007/BF00181355
[8] Glover, D.J.: A study of certain modular representations. J. Algebra51, 425–475 (1978) · Zbl 0376.20008 · doi:10.1016/0021-8693(78)90116-3
[9] Henn, H.-W.: Classifying spaces with injective modp cohomology. Comment. Math. Helv.64, 200–206 (1989) · Zbl 0685.55012 · doi:10.1007/BF02564670
[10] Henn, H.-W., Lannes, J., Schwartz, L.: The categories of unstable modules and unstable algebras over the Steenrod algebra modulo nilpotent objects. Preprint 1989 · Zbl 0805.55011
[11] Higman, D.G.: Finite permutation groups of rank 3. Math. Z.82, 152–175 (1964) · Zbl 0122.03205
[12] Harris, J., Kuhn, N.: Stable decompositions of classifying spaces of finite abelianp-groups. Math. Proc. Camb. Philos. Soc.103, 427–449 (1988) · Zbl 0686.55007 · doi:10.1017/S0305004100065038
[13] Huppert, B.: Endliche Gruppen I. (Grundlagen der mathematischen Wissenschaften vol. 134) Berlin Heidelberg New York: Springer 1967 · Zbl 0217.07201
[14] Huppert, B., Blackburn, N.: Finite groups II, III (Grundlehren der mathematischen Wissenschaften vols 242, 243) Berlin Heidelberg New York: Springer 1982 · Zbl 0514.20002
[15] James, G., Kerber, A.: The representation theory of the symmetric group. Encycl. Math. Appl.16 (1981) · Zbl 0491.20010
[16] Lannes, J.: Sur la cohomologie modulop desp-groupes abéliens élémentaires. Proc. Durham Symp. on Homotopy Theory 1985 (LMS117) Cambridge: Cambridge Univ. Press 1987
[17] Lannes, J., Schwartz, L.: Sur la structure desA-modules instables injectifs. Topology28, 153–169 (1989) · Zbl 0683.55016 · doi:10.1016/0040-9383(89)90018-9
[18] Lannes, J., Zarati, S.: Sur lesU-injectifs. Ann. Sci. Éc. Norm. Supér.19, 303–333 (1986) · Zbl 0608.18006
[19] Lannes, J., Zarati, S.: Sur les foncteurs dérivés de la déstabilisation, avec un appendice de J. Lannes. Math. Z.194, 25–59 (1987) · Zbl 0627.55014 · doi:10.1007/BF01168004
[20] Liebeck, M.: The affine permutation groups of rank three. Proc. Lond. Math. Soc.54, 477–516 (1987) · Zbl 0621.20001 · doi:10.1112/plms/s3-54.3.477
[21] Miller, H.R.: The Sullivan conjecture on maps from classifying spaces. Ann. Math.120, 39–87 (1984) · Zbl 0552.55014 · doi:10.2307/2007071
[22] Mitchell, S.A.: SplittingB(\(\mathbb{Z}\)/p) n andBT n via modular representation theory. Math. Z.189, 1–9 (1985) · Zbl 0547.55017 · doi:10.1007/BF01246939
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.