### Abstract

A compact, connected, simple Lie group G localized at an odd prime p is shown to be homotopy equivalent to a product of homotopy associative, homotopy commutative spaces, provided the rank of G is low. This holds for SU(n), for example, if n <= (p-1)(p-3). The homotopy equivalence is usually just as spaces, not multiplicative spaces. Nevertheless, the strong multiplicative features of the factors can be used to prove useful properties, which after looping can be transferred multiplicatively to Omega G. This is applied to prove useful information about the torsion in the homotopy groups of G, including an upper bound on its exponent.

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

Pages (from-to) | 4511-4535 |

Number of pages | 25 |

Journal | Transactions of the American Mathematical Society |

Volume | 359 |

Issue number | 9 |

Early online date | 17 Apr 2007 |

DOIs | |

Publication status | Published - Sep 2007 |

### Keywords

- Lie group
- exponent
- whitehead product
- H-space
- homotopy-groups
- product spaces
- torsion

### Cite this

*Transactions of the American Mathematical Society*,

*359*(9), 4511-4535. https://doi.org/10.1090/S0002-9947-07-04304-8

**The odd primary H-structure of low rank Lie groups and its application to exponents.** / Theriault, Stephen D.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 359, no. 9, pp. 4511-4535. https://doi.org/10.1090/S0002-9947-07-04304-8

}

TY - JOUR

T1 - The odd primary H-structure of low rank Lie groups and its application to exponents

AU - Theriault, Stephen D

PY - 2007/9

Y1 - 2007/9

N2 - A compact, connected, simple Lie group G localized at an odd prime p is shown to be homotopy equivalent to a product of homotopy associative, homotopy commutative spaces, provided the rank of G is low. This holds for SU(n), for example, if n <= (p-1)(p-3). The homotopy equivalence is usually just as spaces, not multiplicative spaces. Nevertheless, the strong multiplicative features of the factors can be used to prove useful properties, which after looping can be transferred multiplicatively to Omega G. This is applied to prove useful information about the torsion in the homotopy groups of G, including an upper bound on its exponent.

AB - A compact, connected, simple Lie group G localized at an odd prime p is shown to be homotopy equivalent to a product of homotopy associative, homotopy commutative spaces, provided the rank of G is low. This holds for SU(n), for example, if n <= (p-1)(p-3). The homotopy equivalence is usually just as spaces, not multiplicative spaces. Nevertheless, the strong multiplicative features of the factors can be used to prove useful properties, which after looping can be transferred multiplicatively to Omega G. This is applied to prove useful information about the torsion in the homotopy groups of G, including an upper bound on its exponent.

KW - Lie group

KW - exponent

KW - whitehead product

KW - H-space

KW - homotopy-groups

KW - product spaces

KW - torsion

U2 - 10.1090/S0002-9947-07-04304-8

DO - 10.1090/S0002-9947-07-04304-8

M3 - Article

VL - 359

SP - 4511

EP - 4535

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 9

ER -