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Contents
Definition
For the orthogonal group is the group of isometries of a real -dimensional Hilbert space. This is naturally a Lie group. This is canonically isomorphic to the group of 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
Homotopy groups
Proposition
For , , then the canonical inclusion of orthogonal groups
is an (n-1)-equivalence, hence induces an isomorphism on homotopy groups in degrees and a surjection in degree .
Proof
Consider the coset quotient projection
By prop. and by this corollary, the projection is a Serre fibration. Furthermore, example identifies the coset with the n-sphere
Therefore the long exact sequence of homotopy groups of the fiber sequence looks like
Since , this implies that
is an isomorphism and that
is surjective. Hence now the statement follows by induction over .
It follows that the homotopy groups are independent of for (the “stable range”). So if , then . By Bott periodicity we have
In the unstable range for low degrees they instead start out as follows
| | | | | | | | | | | | | | | |
---|
| | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| | 0 | | | | | | | | | | | | | |
| “ | 0 | | | | | | | | | | | | | |
| “ | “ | | | | 0 | | 0 | 0 | | | | | | |
| “ | “ | “ | 0 | | 0 | | | | | | | | | |
| “ | “ | “ | “ | 0 | 0 | | | | | | 0 | | | |
| “ | “ | “ | “ | “ | 0 | | | | | | 0 | | | |
| “ | “ | “ | “ | “ | “ | | | | | | 0 | | | |
| “ | “ | “ | “ | “ | “ | “ | | | | | | | | |
| “ | “ | “ | “ | “ | “ | “ | “ | | | | | | | |
| “ | “ | “ | “ | “ | “ | “ | “ | “ | 0 | | | | | |
| “ | “ | “ | “ | “ | “ | “ | “ | “ | “ | | | | | |
| “ | “ | “ | “ | “ | “ | “ | “ | “ | “ | “ | 0 | | | |
| “ | “ | “ | “ | “ | “ | “ | “ | “ | “ | “ | “ | 0 | | |
| “ | “ | “ | “ | “ | “ | “ | “ | “ | “ | “ | “ | “ | 0 | |
| “ | “ | “ | “ | “ | “ | “ | “ | “ | “ | “ | “ | “ | “ | |
The row can be found using Mimura-Toda 63, using , and that is a -covering space of . The row can be derived from the homotopy groups of as found in Mimura 67. Otherwise the table is given in columns , , and in rows , , by the Encyclopedic Dictionary of Mathematics, Table 6.VII in Appendix A.
Note that the maps
are inclusion of the first summand followed by the map sending and , so that stabilization from to induces multiplication by on (e.g. equations (2.1) and (2.2) in (Tamura 57) and surrounding discussion). The same is also true for .
Homology and cohomology
On the ordinary cohomology of the classifying spaces and
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
Fivebrane group to String group to Spin group to special orthogonal group to orthogonal group.
Given a manifold , lifts of the structure map of the -principal bundle to which the tangent bundle is associated through this tower define, respectively
on .
Coset spaces
Example
The n-spheres are coset spaces of orthogonal groups
For fix a unit vector in . Then its orbit under the defining -action on is clearly the canonical embedding . But precisely the subgroup of that consists of rotations around the axis formed by that unit vector stabilizes it, and that subgroup is isomorphic to , hence .
Example
For , the coset
is called the th real Stiefel manifold of .
Proof
Consider the coset quotient projection
By prop. and by this corollary the projection 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 :
This implies the claim. (Exactness of the sequence says that every element in is in the kernel of zero, hence in the image of 0, hence is 0 itself.)
fivebrane group string group spin group special orthogonal group orthogonal group
group | symbol | universal cover | symbol | higher cover | symbol |
---|
orthogonal group | | Pin group | | Tring group | |
special orthogonal group | | Spin group | | String group | |
Lorentz group | | | | | |
anti de Sitter group | | | | | |
conformal group | | | | | |
Narain group | | | | | |
Poincaré group | | Poincaré spin group | | | |
super Poincaré group | | | | | |
superconformal group | | | | | |
Examples
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 are listed for instance in
-
Alexander Abanov, Homotopy groups of Lie groups 2009 (pdf)
-
M. Mimura and H. Toda, Homotopy Groups of , and , 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 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 and 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 and , 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