group theory

∞-Lie theory

# Contents

## Definition

For $n \in \mathbb{N}$ the orthogonal group is the group of isometries of a real $n$-dimensional Hilbert space. This is naturally a Lie group. This is canonically isomorphic to the group of $n \times n$ orthogonal matrices.

More generally there is a notion of orthogonal group of an inner product space.

The analog for complex Hilbert spaces is the unitary group.

## Properties

### Homotopy groups

The homotopy groups of $O = O(n)$ are for $k \in \mathbb{N}$ and for $n\gt k+1$ (the “stable range”) are

$\array{ \pi_{8k+0}(O) & = \mathbb{Z}_2 \\ \pi_{8k+1}(O) & = \mathbb{Z}_2 \\ \pi_{8k+2}(O) & = 0 \\ \pi_{8k+3}(O) & = \mathbb{Z} \\ \pi_{8k+4}(O) & = 0 \\ \pi_{8k+5}(O) & = 0 \\ \pi_{8k+6}(O) & = 0 \\ \pi_{8k+7}(O) & = \mathbb{Z} } \,.$

In the unstable range for low $n$ they instead start out as follows

$G$$\pi_1$$\pi_2$$\pi_3$$\pi_4$$\pi_5$$\pi_6$$\pi_7$$\pi_8$$\pi_9$$\pi_10$$\pi_11$$\pi_12$
$SO(2)$$\mathbb{Z}$00000000000
$SO(3)$$\mathbb{Z}_2$0$\mathbb{Z}$$\mathbb{Z}_2$$\mathbb{Z}_2$$\mathbb{Z}_{12}$$\mathbb{Z}_{2}$$\mathbb{Z}_{2}$$\mathbb{Z}_{3}$$\mathbb{Z}_{15}$$\mathbb{Z}_{2}$$\mathbb{Z}_{2} \oplus \mathbb{Z}_{2}$
$SO(4)$0$\mathbb{Z} \oplus \mathbb{Z}$$\mathbb{Z}_{2} \oplus \mathbb{Z}_{2}$$\mathbb{Z}_{2} \oplus \mathbb{Z}_{2}$$\mathbb{Z}_{12} \oplus \mathbb{Z}_{12}$$\mathbb{Z}_{2} \oplus \mathbb{Z}_{2}$$\mathbb{Z}_{2} \oplus \mathbb{Z}_{2}$$\mathbb{Z}_{3} \oplus \mathbb{Z}_{3}$$\mathbb{Z}_{15}\oplus \mathbb{Z}_{15}$$\mathbb{Z}_{2}\oplus \mathbb{Z}_{2}$$\mathbb{Z}_{2}^{\oplus 4}$
$SO(5)$$\mathbb{Z}$$\mathbb{Z}_2$$\mathbb{Z}_2$0$\mathbb{Z}$00$\mathbb{Z}_{120}$$\mathbb{Z}_{2}$$\mathbb{Z}_{2} \oplus \mathbb{Z}_{2}$
$SO(6)$0$\mathbb{Z}$0$\mathbb{Z}$00$\mathbb{Z}_{120}\oplus\mathbb{Z}_2$$\mathbb{Z}_{4}$$\mathbb{Z}_{60}$
$SO(7)$00$\mathbb{Z}$00$\mathbb{Z}_{8}$$\mathbb{Z}\oplus\mathbb{Z}_{12}$0
$SO(8)$0$\mathbb{Z} \oplus \mathbb{Z}$$\mathbb{Z}_{2}^{\oplus 3}$$\mathbb{Z}_{2}^{\oplus 3}$$\mathbb{Z}_{24} \oplus \mathbb{Z}_{8}$$\mathbb{Z} \oplus \mathbb{Z}_{2}$0
$SO(9)$$\mathbb{Z}$$\mathbb{Z}_{2}\oplus \mathbb{Z}_{2}$$\mathbb{Z}_{2}\oplus \mathbb{Z}_{2}$$\mathbb{Z}_{8}$$\mathbb{Z}\oplus \mathbb{Z}_{2}$0
$SO(10)$$\mathbb{Z}_{2}$$\mathbb{Z}\oplus \mathbb{Z}_{2}$$\mathbb{Z}_{4}$$\mathbb{Z}$$\mathbb{Z}_{12}$
$SO(11)$$\mathbb{Z}_{2}$$\mathbb{Z}_{2}$$\mathbb{Z}$$\mathbb{Z}_{2}$
$SO(12)$0$\mathbb{Z} \oplus \mathbb{Z}$$\mathbb{Z}\oplus \mathbb{Z}_{2}$

The $SO(6)$ row can be found using Mimura-Toda 63, using $Spin(6) = SU(4)$, and that $Spin(6)$ is a $\mathbb{Z}_2$-covering space of $SO(6)$. The $SO(7)$ row can be derived from the homotopy groups of $Spin(7)$ as found in Mimura 67. Otherwise the table is given in columns $\pi_i$, $i=10,11,12$, and in rows $SO(n)$, $n=8,\ldots,12$, by the Encyclopedic Dictionary of Mathematics?, Table 6.VII in Appendix A.

### Whitehead tower and higher orientation structures

The Whitehead tower of the orthogonal group plays an important role in applications related to quantum physics.

The first steps are

$\cdots \to Fivebrane(n) \to String(n) \to Spin(n) \to SO(n) \to \mathrm{O}(n) \,.$

Fivebrane group to String group to Spin group to special orthogonal group to orthogonal group.

Given a manifold $X$, lifts of the structure map $X \to \mathcal{B}O(n)$ of the $O(n)$-principal bundle to which the tangent bundle is associated through this tower define, respectively

on $X$.

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

groupsymboluniversal coversymbolhigher coversymbol
orthogonal group$\mathrm{O}(n)$Pin group$Pin(n)$Tring group$Tring(n)$
special orthogonal group$SO(n)$Spin group$Spin(n)$String group$String(n)$
Lorentz group$\mathrm{O}(n,1)$$\,$$Spin(n,1)$$\,$$\,$
anti de Sitter group$\mathrm{O}(n,2)$$\,$$Spin(n,2)$$\,$$\,$
Narain group$O(n,n)$
Poincaré group$ISO(n,1)$Poincaré spin group$\widehat {ISO}(n,1)$$\,$$\,$
super Poincaré group$sISO(n,1)$$\,$$\,$$\,$$\,$

## References

Examples of sporadic (exceptional) isogenies from spin groups onto orthogonal groups are discussed in

The homotopy groups of $O(n)$ are listed for instance in

• Alexander Abanov, Homotopy groups of Lie groups 2009 (pdf) (Broken link!)

• M. Mimura and H. Toda, Homotopy Groups of $SU(3)$, $SU(4)$ and $Sp(2)$, J. Math. Kyoto Univ. Volume 3, Number 2 (1963), 217-250. (Euclid)

• M. Mimura, The Homotopy groups of Lie groups of low rank, Math. Kyoto Univ. Volume 6, Number 2 (1967), 131-176. (Euclid)

The ordinary cohomology and ordinary homotopy? of the manifolds $SO(n)$ is discussed in

• Harsh V. Pittie, The integral homology and cohomology rings of SO(n) and Spin(n), Journal of Pure and Applied Algebra Volume 73, Issue 2, 19 August 1991, Pages 105–153 (web)

Revised on November 29, 2014 00:12:49 by David Roberts (58.179.239.189)