group 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

### Compactness

###### Proposition

The orthogonal group $O(n)$ is compact topological space, hence in particular a compact Lie group.

### Homotopy groups

###### Proposition

For $n,k \in \mathbb{N}$, $n \leq k$, then the canonical inclusion of orthogonal groups

$O(n) \hookrightarrow O(k)$

is an (n-1)-equivalence, hence induces an isomorphism on homotopy groups in degrees $\lt n-1$ and a surjection in degree $n-1$.

###### Proof

Consider the coset quotient projection

$O(n) \longrightarrow O(n+1) \longrightarrow O(n+1)/O(n) \,.$

By prop. 1 and by this corollary, the projection $O(n+1)\to O(n+1)/O(n)$ is a Serre fibration. Furthermore, example 1 identifies the coset with the n-sphere

$S^{n}\simeq O(n+1)/O(n) \,.$

Therefore the long exact sequence of homotopy groups of the fiber sequence $O(n)\to O(n+1)\to S^n$ looks like

$\cdots \to \pi_{\bullet+1}(S^n) \longrightarrow \pi_\bullet(O(n)) \longrightarrow \pi_\bullet(O(n+1)) \longrightarrow \pi_\bullet(S^n) \to \cdots$

Since $\pi_{\lt n}(S^n) = 0$, this implies that

$\pi_{\lt n-1}(O(n)) \overset{\simeq}{\longrightarrow} \pi_{\lt n-1}(O(n+1))$

is an isomorphism and that

$\pi_{n-1}(O(n)) \overset{\simeq}{\longrightarrow} \pi_{n-1}(O(n+1))$

is surjective. Hence now the statement follows by induction over $k-n$.

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}$$\mathbb{Z}_{2} \oplus \mathbb{Z}_{2}$
$SO(6)$0$\mathbb{Z}$0$\mathbb{Z}$$\mathbb{Z}_{24}$$\mathbb{Z}_2$$\mathbb{Z}_{120}\oplus\mathbb{Z}_2$$\mathbb{Z}_{4}$$\mathbb{Z}_{60}$
$SO(7)$00$\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(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}_{2}\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.

Note that the maps

$\array{ \pi_3(SO(3)) \longrightarrow \pi_3(SO(4)) \longrightarrow \pi_3(SO(5)) \\ \mathbb{Z}\longrightarrow \mathbb{Z}\oplus \mathbb{Z} \longrightarrow\mathbb{Z} }$

are inclusion of the first summand followed by the map sending $(1,0)\mapsto 2$ and $(0,1)\mapsto 1$, so that stabilization from $SO(3)$ to $SO(5)$ induces multiplication by $2$ on $\pi_3$ (e.g. Tamura 57). The same is also true for $\pi_7(SO(7)) \to \pi_7(SO(8)) \to \pi_7(SO(9))$.

### 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 $X$.

### Coset spaces

###### Example

The n-spheres are coset spaces of orthogonal groups

$S^n \simeq O(n+1)/O(n) \,.$

For fix a unit vector in $\mathbb{R}^{n+1}$. Then its orbit under the defining $O(n+1)$-action on $\mathbb{R}^{n+1}$ is clearly the canonical embedding $S^n \hookrightarrow \mathbb{R}^{n+1}$. But precisely the subgroup of $O(n+1)$ that consists of rotations around the axis formed by that unit vector stabilizes it, and that subgroup is isomorphic to $O(n)$, hence $S^n \simeq O(n+1)/O(n)$.

###### Example

For $n \leq k$, the coset

$V_n(\mathbb{R}^k) \coloneqq O(k)/O(k-n)$

is called the $n$th real Stiefel manifold of $\mathbb{R}^k$.

###### Proposition

The Stiefel manifold $V_n(\mathbb{R}^k)$ (example 2) is (k-n-1)-connected.

###### Proof

Consider the coset quotient projection

$O(k-n) \longrightarrow O(k) \longrightarrow O(k)/O(k-n) = V_n(\mathbb{R}^k) \,.$

By prop. 1 and by this corollary the projection $O(k)\to O(k)/O(k-n)$ is a Serre fibration. Therefore there is induced the long exact sequence of homotopy groups of this fiber sequence, and by prop. 2 it has the following form in degrees bounded by $n$:

$\cdots \to \pi_{\bullet \leq k-n-1}(O(k-n)) \overset{epi}{\longrightarrow} \pi_{\bullet \leq k-n-1}(O(k)) \overset{0}{\longrightarrow} \pi_{\bullet \leq k-n-1}(V_n(\mathbb{R}^k)) \overset{0}{\longrightarrow} \pi_{\bullet-1 \lt k-n-1}(O(k)) \overset{\simeq}{\longrightarrow} \pi_{\bullet-1 \lt k-n-1}(O(k-n)) \to \cdots \,.$

This implies the claim. (Exactness of the sequence says that every element in $\pi_{\bullet \leq n-1}(V_n(\mathbb{R}^k))$ is in the kernel of zero, hence in the image of 0, hence is 0 itself.)

$\cdots\to$ 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)$$\,$$\,$
conformal group$\mathrm{O}(n+1,t+1)$$\,$
Narain group$O(n,n)$
Poincaré group$ISO(n,1)$Poincaré spin group$\widehat {ISO}(n,1)$$\,$$\,$
super Poincaré group$sISO(n,1)$$\,$$\,$$\,$$\,$
superconformal group

## 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)

• 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 homology 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)