nLab orthogonal group

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

$$O(n) \hookrightarrow O(N)$$

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. and by this corollary, the projection $O(n+1)\to O(n+1)/O(n)$ is a Serre fibration. Furthermore, example 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

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

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

It follows that the homotopy groups $\pi_k(O(n))$ are independent of $n$ for $n \gt k + 1$ (the “stable range”). So if $O = \underset{\longrightarrow}{\lim}_n O(n)$, then $\pi_k(O(n)) = \pi_k(O)$. By Bott periodicity we have

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

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

$G$	$\pi_1$	$\pi_2$	$\pi_3$	$\pi_4$	$\pi_5$	$\pi_6$	$\pi_7$	$\pi_8$	$\pi_9$	$\pi_10$	$\pi_11$	$\pi_12$	$\pi_13$	$\pi_14$	$\pi_15$
$SO(2)$	$\mathbb{Z}$	0	0	0	0	0	0	0	0	0	0	0	0	0	0
$SO(3)$	$\mathbb{Z}_2$	0	$\mathbb{Z}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	$\mathbb{Z}_{12}$	$\mathbb{Z}_{2}$	$\mathbb{Z}_{2}$	$\mathbb{Z}_{3}$	$\mathbb{Z}_{15}$	$\mathbb{Z}_{2}$	$\mathbb{Z}_{2}^{\oplus 2}$	$\mathbb{Z}_2\oplus\mathbb{Z}_{12}$	$\mathbb{Z}_2^{\oplus 2}\oplus\mathbb{Z}_{84}$	$\mathbb{Z}_2^{\oplus 2}$
$SO(4)$	“	0	$\mathbb{Z}^{\oplus 2}$	$\mathbb{Z}_{2}^{\oplus 2}$	$\mathbb{Z}_{2}^{\oplus 2}$	$\mathbb{Z}_{12}^{\oplus 2}$	$\mathbb{Z}_{2}^{\oplus 2}$	$\mathbb{Z}_{2}^{\oplus 2}$	$\mathbb{Z}_{3}^{\oplus 2}$	$\mathbb{Z}_{15}^{\oplus 2}$	$\mathbb{Z}_{2}^{\oplus 2}$	$\mathbb{Z}_{2}^{\oplus 4}$	$\mathbb{Z}_2^{\oplus 2}\oplus\mathbb{Z}_{12}^{\oplus 2}$	$\mathbb{Z}_2^{\oplus 4}\oplus\mathbb{Z}_{84}^{\oplus 2}$	$\mathbb{Z}_2^{\oplus 4}$
$SO(5)$	“	“	$\mathbb{Z}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	0	$\mathbb{Z}$	0	0	$\mathbb{Z}_{120}$	$\mathbb{Z}_{2}$	$\mathbb{Z}_{2}^{\oplus 2}$	$\mathbb{Z}_2\oplus\mathbb{Z}_4$	$\mathbb{Z}_{1680}$	$\mathbb{Z}_2$
$SO(6)$	“	“	“	0	$\mathbb{Z}$	0	$\mathbb{Z}$	$\mathbb{Z}_{24}$	$\mathbb{Z}_2$	$\mathbb{Z}_2\oplus\mathbb{Z}_{120}$	$\mathbb{Z}_{4}$	$\mathbb{Z}_{60}$	$\mathbb{Z}_4$	$\mathbb{Z}_2\oplus\mathbb{Z}_{1680}$	$\mathbb{Z}_2\oplus\mathbb{Z}_{72}$
$SO(7)$	“	“	“	“	0	0	$\mathbb{Z}$	$\mathbb{Z}_{2}^{\oplus 2}$	$\mathbb{Z}_{2}^{\oplus 2}$	$\mathbb{Z}_{8}$	$\mathbb{Z}_2\oplus\mathbb{Z}$	0	$\mathbb{Z}_2$	$\mathbb{Z}_2\oplus\mathbb{Z}_8\oplus\mathbb{Z}_{2520}$	$\mathbb{Z}_2^{\oplus 4}$
$SO(8)$	“	“	“	“	“	0	$\mathbb{Z}^{\oplus 2}$	$\mathbb{Z}_{2}^{\oplus 3}$	$\mathbb{Z}_{2}^{\oplus 3}$	$\mathbb{Z}_{8} \oplus \mathbb{Z}_{24}$	$\mathbb{Z}_2 \oplus \mathbb{Z}$	0	$\mathbb{Z}^{\oplus 2}$	$\mathbb{Z}_2\oplus\mathbb{Z}_8\oplus\mathbb{Z}_{120}\oplus\mathbb{Z}_{2520}$	$\mathbb{Z}_2^{\oplus 7}$
$SO(9)$	“	“	“	“	“	“	$\mathbb{Z}$	$\mathbb{Z}_{2}^{\oplus 2}$	$\mathbb{Z}_{2}^{\oplus 2}$	$\mathbb{Z}_{8}$	$\mathbb{Z}_2\oplus \mathbb{Z}$	0	$\mathbb{Z}_2$	$\mathbb{Z}_2\oplus\mathbb{Z}_8$	$\mathbb{Z}_2^{\oplus 3}\oplus\mathbb{Z}$
$SO(10)$	“	“	“	“	“	“	“	$\mathbb{Z}_{2}$	$\mathbb{Z}_2\oplus \mathbb{Z}$	$\mathbb{Z}_{4}$	$\mathbb{Z}$	$\mathbb{Z}_{12}$	$\mathbb{Z}_2$	$\mathbb{Z}_8$	$\mathbb{Z}_2^{\oplus 2}\oplus\mathbb{Z}$
$SO(11)$	“	“	“	“	“	“	“	“	$\mathbb{Z}_{2}$	$\mathbb{Z}_{2}$	$\mathbb{Z}$	$\mathbb{Z}_{2}$	$\mathbb{Z}_2^{\oplus 2}$	$\mathbb{Z}_8$	$\mathbb{Z}_2\oplus\mathbb{Z}$
$SO(12)$	“	“	“	“	“	“	“	“	“	0	$\mathbb{Z}^{\oplus 2}$	$\mathbb{Z}_{2}^{\oplus 2}$	$\mathbb{Z}_2^{\oplus 2}$	$\mathbb{Z}_4\oplus\mathbb{Z}_{24}$	$\mathbb{Z}_2\oplus\mathbb{Z}$
$SO(13)$	“	“	“	“	“	“	“	“	“	“	$\mathbb{Z}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	$\mathbb{Z}_8$	$\mathbb{Z}_2\oplus\mathbb{Z}$
$SO(14)$	“	“	“	“	“	“	“	“	“	“	“	0	$\mathbb{Z}$	$\mathbb{Z}_4$	$\mathbb{Z}$
$SO(15)$	“	“	“	“	“	“	“	“	“	“	“	“	0	$\mathbb{Z}_2$	$\mathbb{Z}$
$SO(16)$	“	“	“	“	“	“	“	“	“	“	“	“	“	0	$\mathbb{Z}^{\oplus 2}$
$SO(17)$	“	“	“	“	“	“	“	“	“	“	“	“	“	“	$\mathbb{Z}$

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 $\pi_k$, $k=10,\ldots, 15$, and in rows $SO(n)$, $n=8,\ldots,17$, 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. equations (2.1) and (2.2) in (Tamura 57) and surrounding discussion). The same is also true for $\pi_7(SO(7)) \to \pi_7(SO(8)) \to \pi_7(SO(9))$.

Homology and cohomology

On the ordinary cohomology of the classifying spaces $B O(n)$ and $B S O(n)$

• with $\mathbb{Z}_2$ coefficients:

  generated by the Stiefel-Whitney classes

  (e.g. Milnor & Stasheff 74, Theorem 7.1. & Theorem 12.4.)

• with integer coefficients (integral cohomology):

  generated by the integral Stiefel-Whitney classes and the Pontrjagin classes

  (e.g. Brown 82, Feshbach 83, Pittie 91, Rudolph-Schmidt 17, Theorem 4.2.23)

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

• orientation

• Spin structure

• String structure

• Fivebrane structure

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 ) 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. 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. 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.)

Related concepts

• orthogonal calculus

• Stiefel manifold, Grassmannian

• stable orthogonal group

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

group	symbol	universal cover	symbol	higher cover	symbol
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

Examples

rotation groups in low dimensions:

Dynkin label	sp. orth. group	spin group	pin group	semi-spin group
	SO(2)	Spin(2)	Pin(2)
B1	SO(3)	Spin(3)	Pin(3)
D2	SO(4)	Spin(4)	Pin(4)
B2	SO(5)	Spin(5)	Pin(5)
D3	SO(6)	Spin(6)
B3	SO(7)	Spin(7)
D4	SO(8)	Spin(8)		SO(8)
B4	SO(9)	Spin(9)
D5	SO(10)	Spin(10)
B5	SO(11)	Spin(11)
D6	SO(12)	Spin(12)
	$\vdots$	$\vdots$
D8	SO(16)	Spin(16)		SemiSpin(16)
	$\vdots$	$\vdots$
D16	SO(32)	Spin(32)		SemiSpin(32)

see also

• Spin(5).Spin(3)

• finite rotation groups

• ADE-classification

References

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

• Paul Garrett, Sporadic isogenies to orthogonal groups (July 2013) [pdf, pdf ]

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

• John Milnor, Jim Stasheff, Characteristic classes, Princeton Univ. Press (1974) [ISBN:9780691081229, doi:10.1515/9781400881826, pdf]

• Edgar H. Brown, The Cohomology of $B SO_n$ and $BO_n$ with Integer Coefficients, Proceedings of the American Mathematical Society Vol. 85, No. 2 (Jun., 1982), pp. 283-288 (jstor:2044298)

• Mark Feshbach, The Integral Cohomology Rings of the Classifying Spaces of $\mathrm{O}(n)$ and $\mathrm{SO}(n)$, Indiana Univ. Math. J. 32 (1983), 511-516 (doi:10.1512/iumj.1983.32.32036)

• 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 (doi:10.1016/0022-4049(91)90108-E))

• Gerd Rudolph, Matthias Schmidt, around Theorem 4.2.23 of Differential Geometry and Mathematical Physics: Part II. Fibre Bundles, Topology and Gauge Fields, Theoretical and Mathematical Physics series, Springer 2017 (doi:10.1007/978-94-024-0959-8)

See also

• Itiro Tamura, On Pontrjagin classes of homotopy types of manifolds, Journal of the mathematical society of Japan, Vol. 9 No. 2 , 1957 pdf