Contents

# Contents

## Definition

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.

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

## Properties

### Homotopy groups

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

$n\;mod\; 8$01234567
$\pi_n(O)$$\mathbb{Z}_2$$\mathbb{Z}_2$0$\mathbb{Z}$000$\mathbb{Z}$

or if we instead write down the order:

$n\;mod\; 8$01234567
${\vert\pi_n(O)\vert}$221$\infty$111$\infty$

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

$n$012345678910111213141516
homotopy groups of stable orthogonal group$\pi_n(O)$$\mathbb{Z}_2$$\mathbb{Z}_2$0$\mathbb{Z}$000$\mathbb{Z}$$\mathbb{Z}_2$$\mathbb{Z}_2$0$\mathbb{Z}$000$\mathbb{Z}$$\mathbb{Z}_2$
stable homotopy groups of spheres$\pi_n(\mathbb{S})$$\mathbb{Z}$$\mathbb{Z}_2$$\mathbb{Z}_2$$\mathbb{Z}_{24}$00$\mathbb{Z}_2$$\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$$\mathbb{Z}_2 \oplus \mathbb{Z}_2$
image of J-homomorphism$im(\pi_n(J))$0$\mathbb{Z}_2$0$\mathbb{Z}_{24}$000$\mathbb{Z}_{240}$$\mathbb{Z}_2$$\mathbb{Z}_2$0$\mathbb{Z}_{504}$000$\mathbb{Z}_{480}$$\mathbb{Z}_2$