Cohomology and Extensions
∞-Lie theory (higher geometry)
Formal Lie groupoids
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.
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 .
Consider the coset quotient projection
By prop. 1 and by this corollary, the projection is a Serre fibration. Furthermore, example 1 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 .
The homotopy groups of are for and for (the “stable range”) are
In the unstable range for low they instead start out as follows
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.
Beware that the maps
are inclusion of the first summand followed by projection onto the second. So even though superficially looks like it already stabilized, in fact these elements disappear in the stabilization and another copy of appears (e.g. Tamura 57). The same is also true for .
Homology and cohomology
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
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 .
For , the coset
is called the th real Stiefel manifold of .
Consider the coset quotient projection
By prop. 1 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. 2 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|
|anti de Sitter group|
|Poincaré group||Poincaré spin group|
|super Poincaré group|
Examples of sporadic (exceptional) isogenies from spin groups onto orthogonal groups are discussed in
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
- 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)
- Itiro Tamura, On Pontrjagon classes of homotopy types of manifolds, Journal of the mathematical society of Japan, Vol. 9 No. 2 , 1957 pdf