The -fold product of an arbitrary space usually supports only the obvious permutation action of the symmetric group . However, if is a -complete, homotopy associative, homotopy commutative -space one can define a homotopy action of on . In various cases, e.g. if multiplication by is null homotopic then we get a homotopy action of for some . After one suspension this allows one to split using idempotents of which can be lifted to . In fact all of this is possible if is an -space whose homology algebra is commutative and nilpotent. For we make some explicit calculations of splittings of , ,and .
Habituellement le produit de copies d’un espace arbitraire ne soutient que l’action de permutation du groupe symétrique . Cependant, si est un -espace, - complet, associatif et commutatif à homotopie près on peut définir une action à homotopie près de sur . Dans divers cas, par exemple, si la multiplication par est nulle homotopique, on obtient une action à homotopie près de pour certains . Après une suspension cela permet de décomposer en utilisant des idempotents de qui peuvent être relevés sur . En fait, tout ceci est possible si est un -espace pour lequel l’algèbre est commutative et nilpotente. Pour nous faisons des calculs explicites de décomposition de , ,et .
Keywords: splittings, $H$-spaces
Mot clés : décompositions, $H$-espaces
@article{AIF_2001__51_6_1719_0, author = {Levi, Ran and Priddy, Stewart}, title = {On certain homotopy actions of general linear groups on iterated products}, journal = {Annales de l'Institut Fourier}, pages = {1719--1739}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {51}, number = {6}, year = {2001}, doi = {10.5802/aif.1872}, mrnumber = {1871287}, zbl = {0990.55003}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.1872/} }
TY - JOUR AU - Levi, Ran AU - Priddy, Stewart TI - On certain homotopy actions of general linear groups on iterated products JO - Annales de l'Institut Fourier PY - 2001 SP - 1719 EP - 1739 VL - 51 IS - 6 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.1872/ DO - 10.5802/aif.1872 LA - en ID - AIF_2001__51_6_1719_0 ER -
%0 Journal Article %A Levi, Ran %A Priddy, Stewart %T On certain homotopy actions of general linear groups on iterated products %J Annales de l'Institut Fourier %D 2001 %P 1719-1739 %V 51 %N 6 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.1872/ %R 10.5802/aif.1872 %G en %F AIF_2001__51_6_1719_0
Levi, Ran; Priddy, Stewart. On certain homotopy actions of general linear groups on iterated products. Annales de l'Institut Fourier, Volume 51 (2001) no. 6, pp. 1719-1739. doi : 10.5802/aif.1872. http://archive.numdam.org/articles/10.5802/aif.1872/
[1] Polynomial Invariants of Finite Groups, L.M.S. Lecture Notes in Mathematics, Volume 190 (1993) | MR | Zbl
[2] Homotopy Limits, Completions and Localizations, Springer Lecture Notes in Mathematics, Volume 304 (1972) | DOI | MR | Zbl
[3] Nilpotence and stable homotopy theory. I, Ann. of Math. (2), Volume 128 (1988), pp. 207-241 | DOI | MR | Zbl
[4] Stable decompositions of classifying spaces of finite abelian -groups, Math. Proc. Camb. Phil. Soc., Volume 103 (1988), pp. 427-449 | DOI | MR | Zbl
[5] The transfer and Whitehead's conjecture, Math. Proc. Cambridge Philos. Soc., Volume 98 (1985), pp. 459-480 | DOI | MR | Zbl
[6] A new infinite family in , Topology, Volume 16 (1977), pp. 249-256 | DOI | MR | Zbl
[7] On the Steinberg module, representations of the symmetric groups, and the Steenrod algebra, J. Pure Appl. Algebra, Volume 39 (1986), pp. 275-281 | DOI | MR | Zbl
[8] Finite complexes with -free cohomology, Topology, Volume 24 (1985), pp. 227-246 | DOI | MR | Zbl
[9] Stable splittings derived from the Steinberg module, Topology, Volume 22 (1983), pp. 219-232 | MR | Zbl
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