### 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 CLn(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 GLn(Z/(T)(p)) 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/(r)(p)). In fact all of this is possible if X is an H-space whose homology algebra H-* (X; Z/(p)) is commutative and nilpotent. For n = 2 we make some explicit calculations of splittings of Sigma(SO (4) xSO(4)), Sigma(Omega(2)S(3) XOmega(2)S(3)) and Sigma(G(2) x G(2)).

Original language | English |
---|---|

Pages (from-to) | 1719-1739 |

Number of pages | 20 |

Journal | Annales de l'Institut Fourier |

Volume | 51 |

Issue number | 6 |

Publication status | Published - 2001 |

### Keywords

- splittings
- H-spaces

### Cite this

*Annales de l'Institut Fourier*,

*51*(6), 1719-1739.

**On certain homotopy actions of general linear groups on iterated products.** / Levi, Ran; Priddy, S.

Research output: Contribution to journal › Article

*Annales de l'Institut Fourier*, vol. 51, no. 6, pp. 1719-1739.

}

TY - JOUR

T1 - On certain homotopy actions of general linear groups on iterated products

AU - Levi, Ran

AU - Priddy, S.

PY - 2001

Y1 - 2001

N2 - 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 CLn(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 GLn(Z/(T)(p)) 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/(r)(p)). In fact all of this is possible if X is an H-space whose homology algebra H-* (X; Z/(p)) is commutative and nilpotent. For n = 2 we make some explicit calculations of splittings of Sigma(SO (4) xSO(4)), Sigma(Omega(2)S(3) XOmega(2)S(3)) and Sigma(G(2) x G(2)).

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 CLn(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 GLn(Z/(T)(p)) 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/(r)(p)). In fact all of this is possible if X is an H-space whose homology algebra H-* (X; Z/(p)) is commutative and nilpotent. For n = 2 we make some explicit calculations of splittings of Sigma(SO (4) xSO(4)), Sigma(Omega(2)S(3) XOmega(2)S(3)) and Sigma(G(2) x G(2)).

KW - splittings

KW - H-spaces

M3 - Article

VL - 51

SP - 1719

EP - 1739

JO - Annales de l'Institut Fourier

JF - Annales de l'Institut Fourier

SN - 0373-0956

IS - 6

ER -