On certain homotopy actions of general linear groups on iterated products
Annales de l'Institut Fourier, Volume 51 (2001) no. 6, p. 1719-1739

The n-fold product X n of an arbitrary space usually supports only the obvious permutation action of the symmetric group Σ n . However, if X is a p-complete, homotopy associative, homotopy commutative H-space one can define a homotopy action of GL n ( p ) on X n . In various cases, e.g. if multiplication by p r is null homotopic then we get a homotopy action of GL n (/p r ) for some r. After one suspension this allows one to split X n using idempotents of 𝔽 p GL n (/p) which can be lifted to 𝔽 p GL n (/p r ). In fact all of this is possible if X is an H-space whose homology algebra H * (X;BbbZ/p) is commutative and nilpotent. For n=2 we make some explicit calculations of splittings of Σ( SO (4)× SO (4)), Σ(Ω 2 S 3 ×Ω 2 S 3 ),and Σ(G 2 ×G 2 ).

Habituellement le produit de n copies d’un espace arbitraire ne soutient que l’action de permutation du groupe symétrique Σ n . Cependant, si X est un H-espace, p- complet, associatif et commutatif à homotopie près on peut définir une action à homotopie près de GL n ( p ) sur X n . Dans divers cas, par exemple, si la multiplication par p r est nulle homotopique, on obtient une action à homotopie près de GL n (/p r ) pour certains r. Après une suspension cela permet de décomposer X n en utilisant des idempotents de 𝔽 p GL n (/p) qui peuvent être relevés sur BbbF p GL n (/p r ). En fait, tout ceci est possible si X est un H-espace pour lequel l’algèbre H * (X;/p) est commutative et nilpotente. Pour n=2 nous faisons des calculs explicites de décomposition de Σ( SO (4)× SO (4)), Σ(Ω 2 S 3 ×Ω 2 S 3 ),et Σ(G 2 ×G 2 ).

DOI : https://doi.org/10.5802/aif.1872
Classification:  55P45,  55R35,  20C20
Keywords: splittings, H-spaces
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     author = {Levi, Ran and Priddy, Stewart},
     title = {On certain homotopy actions of general linear groups on iterated products},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {51},
     number = {6},
     year = {2001},
     pages = {1719-1739},
     doi = {10.5802/aif.1872},
     zbl = {0990.55003},
     mrnumber = {1871287},
     language = {en},
     url = {http://www.numdam.org/item/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://www.numdam.org/item/AIF_2001__51_6_1719_0/

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