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

Stephen D Theriault

Research output: Contribution to journalArticle

23 Citations (Scopus)

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 languageEnglish
Pages (from-to)4511-4535
Number of pages25
JournalTransactions of the American Mathematical Society
Volume359
Issue number9
Early online date17 Apr 2007
DOIs
Publication statusPublished - Sep 2007

Keywords

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

Cite this

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

In: Transactions of the American Mathematical Society, Vol. 359, No. 9, 09.2007, p. 4511-4535.

Research output: Contribution to journalArticle

@article{0bbbb5aaedda438d8f2947c7580f7579,
title = "The odd primary H-structure of low rank Lie groups and its application to exponents",
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.",
keywords = "Lie group, exponent, whitehead product, H-space, homotopy-groups, product spaces, torsion",
author = "Theriault, {Stephen D}",
year = "2007",
month = "9",
doi = "10.1090/S0002-9947-07-04304-8",
language = "English",
volume = "359",
pages = "4511--4535",
journal = "Transactions of the American Mathematical Society",
issn = "0002-9947",
publisher = "American Mathematical Society",
number = "9",

}

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 -