# On certain homotopy actions of general linear groups on iterated products

Ran Levi^{[1]}; Stewart Priddy^{[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 (2001)

- Volume: 51, Issue: 6, page 1719-1739
- ISSN: 0373-0956

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topLevi, Ran, and Priddy, Stewart. "On certain homotopy actions of general linear groups on iterated products." Annales de l’institut Fourier 51.6 (2001): 1719-1739. <http://eudml.org/doc/115965>.

@article{Levi2001,

abstract = {The $n$-fold product $X^n$ of an arbitrary space usually supports only the obvious
permutation action of the symmetric group $\Sigma _n$. However, if $X$ is a $p$-complete,
homotopy associative, homotopy commutative $H$-space one can define a homotopy action of
$\{\rm GL\}_n(\{\mathbb \{Z\}\}_p)$ on $X^n$. In various cases, e.g. if multiplication by $p^r$ is
null homotopic then we get a homotopy action of $\{\rm G\}L_n(\{\mathbb \{Z\}\}/p^r)$ for some $r$.
After one suspension this allows one to split $X^n$ using idempotents of $\{\mathbb \{F\}\}_p\{\rm GL\}_n(\{\mathbb \{Z\}\}/p)$ which can be lifted to $\{\mathbb \{F\}\}_p\{\rm GL\}_n(\{\mathbb \{Z\}\}/p^r)$. In fact
all of this is possible if $X$ is an $H$-space whose homology algebra $H_*(X;\{Bbb Z\}/p)$ is commutative and nilpotent. For $n=2$ we make some explicit calculations of splittings
of $\Sigma (\{\rm SO\}(4)\times \{\rm SO\}(4))$, $\Sigma (\Omega ^2 S^3 \times \Omega ^2 S^3)$,and
$\Sigma (G_2 \times G_2)$.},

affiliation = {University of Aberdeen, Department of Mathematics, Aberdeen (Grande-Bretagne); Northwestern University, Department of Mathematics, Evanston IL 60208 (USA)},

author = {Levi, Ran, Priddy, Stewart},

journal = {Annales de l’institut Fourier},

keywords = {splittings; $H$-spaces; -spaces; decompositions},

language = {eng},

number = {6},

pages = {1719-1739},

publisher = {Association des Annales de l'Institut Fourier},

title = {On certain homotopy actions of general linear groups on iterated products},

url = {http://eudml.org/doc/115965},

volume = {51},

year = {2001},

}

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

PB - Association des Annales de l'Institut Fourier

VL - 51

IS - 6

SP - 1719

EP - 1739

AB - The $n$-fold product $X^n$ of an arbitrary space usually supports only the obvious
permutation action of the symmetric group $\Sigma _n$. However, if $X$ is a $p$-complete,
homotopy associative, homotopy commutative $H$-space one can define a homotopy action of
${\rm GL}_n({\mathbb {Z}}_p)$ on $X^n$. In various cases, e.g. if multiplication by $p^r$ is
null homotopic then we get a homotopy action of ${\rm G}L_n({\mathbb {Z}}/p^r)$ for some $r$.
After one suspension this allows one to split $X^n$ using idempotents of ${\mathbb {F}}_p{\rm GL}_n({\mathbb {Z}}/p)$ which can be lifted to ${\mathbb {F}}_p{\rm GL}_n({\mathbb {Z}}/p^r)$. In fact
all of this is possible if $X$ is an $H$-space whose homology algebra $H_*(X;{Bbb Z}/p)$ is commutative and nilpotent. For $n=2$ we make some explicit calculations of splittings
of $\Sigma ({\rm SO}(4)\times {\rm SO}(4))$, $\Sigma (\Omega ^2 S^3 \times \Omega ^2 S^3)$,and
$\Sigma (G_2 \times G_2)$.

LA - eng

KW - splittings; $H$-spaces; -spaces; decompositions

UR - http://eudml.org/doc/115965

ER -

## References

top- D. Benson, Polynomial Invariants of Finite Groups, L.M.S. Lecture Notes in Mathematics 190 (1993) Zbl0864.13001MR1233169
- A.K. Bousfield, D. Kan, Homotopy Limits, Completions and Localizations, Springer Lecture Notes in Mathematics 304 (1972) Zbl0259.55004MR365573
- E. Devinatz, J. Smith, M. Hopkins, Nilpotence and stable homotopy theory. I, Ann. of Math. (2) 128 (1988), 207-241 Zbl0673.55008MR960945
- J. Harris, N. Kuhn, Stable decompositions of classifying spaces of finite abelian $p$-groups, Math. Proc. Camb. Phil. Soc. 103 (1988), 427-449 Zbl0686.55007MR932667
- N. Kuhn, S. Priddy, The transfer and Whitehead's conjecture, Math. Proc. Cambridge Philos. Soc. 98 (1985), 459-480 Zbl0584.55007MR803606
- M. Mahowald, A new infinite family in ${}_{2}{\pi}_{*}^{s}$, Topology 16 (1977), 249-256 Zbl0357.55020MR445498
- S. Mitchell, On the Steinberg module, representations of the symmetric groups, and the Steenrod algebra, J. Pure Appl. Algebra 39 (1986), 275-281 Zbl0593.20006MR821892
- S. Mitchell, Finite complexes with $A\left(n\right)$-free cohomology, Topology 24 (1985), 227-246 Zbl0568.55021MR793186
- S. Mitchell, S. Priddy, Stable splittings derived from the Steinberg module, Topology 22 (1983), 219-232 Zbl0526.55010MR710102

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