Indefinite Orthogonal Group - Topology

Topology

Assuming both p and q are nonzero, neither of the groups O(p,q) or SO(p,q) are connected, having four and two components respectively. π₀(O(p,q)) ≅ C₂ × C₂ is the Klein four-group, with each factor being whether an element preserves or reverses the respective orientations on the p and q dimensional subspaces on which the form is definite; note that reserving orientation on only one of these subspaces reverses orientation on the whole space. The special orthogonal group has components π₀(SO(p,q)) = {(1,1),(−1,−1)} which either preserves both orientations or reverses both orientations, in either case preserving the overall orientation.

The identity component of O(p,q) is often denoted SO+(p,q) and can be identified with the set of elements in SO(p,q) which preserves both orientations. This notation is related to the notation O+(1,3) for the orthochronous Lorentz group, where the + refers to preserving the orientation on the first (temporal) dimension.

The group O(p,q) is also not compact, but contains the compact subgroups O(p) and O(q) acting on the subspaces on which the form is definite. In fact, O(p) × O(q) is a maximal compact subgroup of O(p,q), while S(O(p) × O(q)) is a maximal compact subgroup of SO(p,q). Likewise, SO(p) × SO(q) is a maximal compact subgroup of SO+(p, q). Thus up to homotopy, the spaces are products of (special) orthogonal groups, from which algebro-topological invariants can be computed.

In particular, the fundamental group of SO+(p,q) is the product of the fundamental groups of the components, π₁(SO+(p,q)) = π₁(SO(p)) × π₁(SO(q)), and is given by:

π1(SO+(p,q))	p = 1	p = 2	p ≥ 3
q = 1	{1}	Z	Z2
q = 2	Z	Z × Z	Z × Z2
q ≥ 3	Z2	Z2 × Z	Z2 × Z2