nLab stable orthogonal group

Contents

Context

Group Theory

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Definition

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

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

of the orthogonal group in dimension nn into that in dimension n+1n+1. The stable orthogonal group is the direct limit (as topological groups of this sequence of inclusions.

Olim nO(n). 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

nmod8n\;mod\; 801234567
π n(O)\pi_n(O) 2\mathbb{Z}_2 2\mathbb{Z}_20\mathbb{Z}000\mathbb{Z}

or if we instead write down the order:

nmod8n\;mod\; 801234567
|π n(O)|{\vert\pi_n(O)\vert}221\infty111\infty

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

nn012345678910111213141516
Whitehead tower of orthogonal grouporientationspin groupstring groupfivebrane groupninebrane group
higher versionsspecial orthogonal groupspin groupstring 2-groupfivebrane 6-groupninebrane 10-group
homotopy groups of stable orthogonal groupπ n(O)\pi_n(O) 2\mathbb{Z}_2 2\mathbb{Z}_20\mathbb{Z}000\mathbb{Z} 2\mathbb{Z}_2 2\mathbb{Z}_20\mathbb{Z}000\mathbb{Z} 2\mathbb{Z}_2
stable homotopy groups of spheresπ n(𝕊)\pi_n(\mathbb{S})\mathbb{Z} 2\mathbb{Z}_2 2\mathbb{Z}_2 24\mathbb{Z}_{24}00 2\mathbb{Z}_2 240\mathbb{Z}_{240} 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 2 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2 6\mathbb{Z}_6 504\mathbb{Z}_{504}0 3\mathbb{Z}_3 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 480 2\mathbb{Z}_{480} \oplus \mathbb{Z}_2 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2
image of J-homomorphismim(π n(J))im(\pi_n(J))0 2\mathbb{Z}_20 24\mathbb{Z}_{24}000 240\mathbb{Z}_{240} 2\mathbb{Z}_2 2\mathbb{Z}_20 504\mathbb{Z}_{504}000 480\mathbb{Z}_{480} 2\mathbb{Z}_2

Last revised on March 11, 2024 at 05:24:35. See the history of this page for a list of all contributions to it.