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

Abstract

top
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 ) × SO ( 4 ) ) , Σ ( Ω 2 S 3 × Ω 2 S 3 ) ,and Σ ( G 2 × G 2 ) .

How to cite

top

Levi, 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
  1. D. Benson, Polynomial Invariants of Finite Groups, L.M.S. Lecture Notes in Mathematics 190 (1993) Zbl0864.13001MR1233169
  2. A.K. Bousfield, D. Kan, Homotopy Limits, Completions and Localizations, Springer Lecture Notes in Mathematics 304 (1972) Zbl0259.55004MR365573
  3. E. Devinatz, J. Smith, M. Hopkins, Nilpotence and stable homotopy theory. I, Ann. of Math. (2) 128 (1988), 207-241 Zbl0673.55008MR960945
  4. 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
  5. N. Kuhn, S. Priddy, The transfer and Whitehead's conjecture, Math. Proc. Cambridge Philos. Soc. 98 (1985), 459-480 Zbl0584.55007MR803606
  6. M. Mahowald, A new infinite family in 2 π * s , Topology 16 (1977), 249-256 Zbl0357.55020MR445498
  7. 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
  8. S. Mitchell, Finite complexes with A ( n ) -free cohomology, Topology 24 (1985), 227-246 Zbl0568.55021MR793186
  9. S. Mitchell, S. Priddy, Stable splittings derived from the Steinberg module, Topology 22 (1983), 219-232 Zbl0526.55010MR710102

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.