stable orthogonal group

For each $n \in \mathbb{N}$ there is an inclusion

$O(n) \hookrightarrow O(n+1)$

of the orthogonal group in dimension $n$ into that in dimension $n+1$. The *stable orthogonal group* is the direct limit over this sequence of inclusions.

$O \coloneqq {\underset{\to}{\lim}}_n O(n)
\,.$

By the discussion at *orthogonal group – homotopy groups* we have that the homotopy groups of the stable orthogonal group are

$n\;mod\; 8$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|

$\pi_n(O)$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | 0 | $\mathbb{Z}$ | 0 | 0 | 0 | $\mathbb{Z}$ |

or if we instead write down the order:

$n\;mod\; 8$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|

${\vert\pi_n(O)\vert}$ | 2 | 2 | 1 | $\infty$ | 1 | 1 | 1 | $\infty$ |

Via the J-homomorphism this is related to the stable homotopy groups of spheres:

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

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

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 | $\mathbb{Z}_{480}$ | $\mathbb{Z}_2$ |

Revised on July 14, 2017 05:16:51
by Urs Schreiber
(178.6.113.200)