The low degrees in the Whitehead tower of the orthogonal group are closely linked to the physics of branes appearing in string theory/M-theory. Accordingly the low stages are called

$\cdots \to$ fivebrane group $\to$ string group $\to$ spin group $\to$ special orthogonal group $\to$ orthogonal group

To which extent and in which sense this matching with brane physics continues is not entirely clear. Hisham Sati gives arguments that co-killing the next integer homotopy group $\pi_{11}(O)$ corresponds to the M9-brane. Therefore one might call the corresponding stage in the Whitehead tower the *ninebrane group*. This suggestion appeared in (Sati-Schreiber-Stasheff 08)

$n$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Whitehead tower of orthogonal group | orientation | spin group | string group | fivebrane group | ninebrane group | |||||||||||||

higher versions | special orthogonal group | spin group | string 2-group | fivebrane 6-group | ninebrane 10-group | |||||||||||||

homotopy groups of stable orthogonal group | $\pi_n(O)$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | 0 | $\mathbb{Z}$ | 0 | 0 | 0 | $\mathbb{Z}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | 0 | $\mathbb{Z}$ | 0 | 0 | 0 | $\mathbb{Z}$ | $\mathbb{Z}_2$ |

stable homotopy groups of spheres | $\pi_n(\mathbb{S})$ | $\mathbb{Z}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | $\mathbb{Z}_{24}$ | 0 | 0 | $\mathbb{Z}_2$ | $\mathbb{Z}_{240}$ | $\mathbb{Z}_2 \oplus \mathbb{Z}_2$ | $\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$ | $\mathbb{Z}_6$ | $\mathbb{Z}_{504}$ | 0 | $\mathbb{Z}_3$ | $\mathbb{Z}_2 \oplus \mathbb{Z}_2$ | $\mathbb{Z}_{480} \oplus \mathbb{Z}_2$ | $\mathbb{Z}_2 \oplus \mathbb{Z}_2$ |

image of J-homomorphism | $im(\pi_n(J))$ | 0 | $\mathbb{Z}_2$ | 0 | $\mathbb{Z}_{24}$ | 0 | 0 | 0 | $\mathbb{Z}_{240}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | 0 | $\mathbb{Z}_{504}$ | 0 | 0 | 0 | $\mathbb{Z}_{480}$ | $\mathbb{Z}_2$ |

- Hisham Sati, Urs Schreiber, Jim Stasheff,
*L-∞ algebra connections*, in -Quantum Field Theory_, Birkhäuser (2009), 303-424, DOI: 10.1007/978-3-7643-8736-5_17 (arXiv:0801.3480)

Last revised on May 27, 2022 at 02:52:10. See the history of this page for a list of all contributions to it.