On certain homotopy actions of general linear groups on iterated products

Levi, Ran; Priddy, Stewart

doi:10.5802/aif.1872

On certain homotopy actions of general linear groups on iterated products
[Sur certaines actions d'homotopie des groupes linéaires généraux sur des produits itérés]

Levi, Ran ¹ ; Priddy, Stewart ²

¹ University of Aberdeen, Department of Mathematics, Aberdeen (Grande-Bretagne)
² Northwestern University, Department of Mathematics, Evanston IL 60208 (USA)

Annales de l'Institut Fourier, Tome 51 (2001) no. 6, pp. 1719-1739.

Résumé
Abstract

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 ${GL}_{n} (ℤ_{p})$ 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 ${GL}_{n} (ℤ / p^{r})$ pour certains $r$ . Après une suspension cela permet de décomposer $X^{n}$ en utilisant des idempotents de $𝔽_{p} {GL}_{n} (ℤ / p)$ qui peuvent être relevés sur ${B b b F}_{p} {GL}_{n} (ℤ / p^{r})$ . En fait, tout ceci est possible si $X$ est un $H$ -espace pour lequel l’algèbre $H_{*} (X; ℤ / p)$ est commutative et nilpotente. Pour $n = 2$ nous faisons des calculs explicites de décomposition de $Σ (SO (4) \times SO (4))$ , $Σ (Ω^{2} S^{3} \times Ω^{2} S^{3})$ ,et $Σ (G_{2} \times G_{2})$ .

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 ${GL}_{n} (ℤ_{p})$ on $X^{n}$ . In various cases, e.g. if multiplication by $p^{r}$ is null homotopic then we get a homotopy action of $G L_{n} (ℤ / p^{r})$ for some $r$ . After one suspension this allows one to split $X^{n}$ using idempotents of $𝔽_{p} {GL}_{n} (ℤ / p)$ which can be lifted to $𝔽_{p} {GL}_{n} (ℤ / p^{r})$ . In fact all of this is possible if $X$ is an $H$ -space whose homology algebra $H_{*} (X; B b b Z / p)$ is commutative and nilpotent. For $n = 2$ we make some explicit calculations of splittings of $Σ (SO (4) \times SO (4))$ , $Σ (Ω^{2} S^{3} \times Ω^{2} S^{3})$ ,and $Σ (G_{2} \times G_{2})$ .

MR Zbl

DOI : 10.5802/aif.1872

Classification : 55P45, 55R35, 20C20
Keywords: splittings,

H

-spaces
Mot clés : décompositions,

H

-espaces

Affiliations des auteurs :

Levi, Ran ¹ ; Priddy, Stewart ²

¹ University of Aberdeen, Department of Mathematics, Aberdeen (Grande-Bretagne)
² Northwestern University, Department of Mathematics, Evanston IL 60208 (USA)

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     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 = {https://www.numdam.org/articles/10.5802/aif.1872/}
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Levi, Ran; Priddy, Stewart. On certain homotopy actions of general linear groups on iterated products. Annales de l'Institut Fourier, Tome 51 (2001) no. 6, pp. 1719-1739. doi : 10.5802/aif.1872. https://www.numdam.org/articles/10.5802/aif.1872/

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