Editing Orthogonal group (section)

==== Computation and interpretation of homotopy groups ====

===== Low-dimensional groups =====
The first few homotopy groups can be calculated by using the concrete descriptions of low-dimensional groups.
* {{math|1=π<sub>0</sub>(''O'') = π<sub>0</sub>(O(1)) = '''Z''' / 2'''Z'''}}, from [[orientation (mathematics)|orientation]]-preserving/reversing (this class survives to {{math|O(2)}} and hence stably)
* {{math|1=π<sub>1</sub>(''O'') = π<sub>1</sub>(SO(3)) = '''Z''' / 2'''Z'''}}, which is [[spin group|spin]] comes from {{math|1=SO(3) = '''R'''P<sup>3</sup> = ''S''<sup>3</sup> / ('''Z''' / 2'''Z''')}}.
* {{math|1=π<sub>2</sub>(''O'') = π<sub>2</sub>(SO(3)) = 0}}, which surjects onto {{math|π<sub>2</sub>(SO(4))}}; this latter thus vanishes.

===== Lie groups =====
From general facts about [[Lie group]]s, {{math|π<sub>2</sub>(''G'')}} always vanishes, and {{math|π<sub>3</sub>(''G'')}} is free ([[free abelian group|free abelian]]).

===== Vector bundles =====
{{confusing section|date=January 2024}}
{{math|π<sub>0</sub>(''K''O)}} is a [[vector bundle]] over {{math|''S''<sup>0</sup>}}, which consists of two points. Thus over each point, the bundle is trivial, and the non-triviality of the bundle is the difference between the dimensions of the vector spaces over the two points, so {{math|1=π<sub>0</sub>(''K''O) = '''[[integers|Z]]'''}} is the [[Hamel dimension|dimension]].

===== Loop spaces =====
Using concrete descriptions of the loop spaces in [[Bott periodicity]], one can interpret the higher homotopies of {{math|''O''}} in terms of simpler-to-analyze homotopies of lower order. Using π<sub>0</sub>, {{math|''O''}} and {{math|''O''/U}} have two components, {{math|1=''K''O = ''B''O × '''Z'''}} and {{math|1=''K''Sp = ''B''Sp × '''Z'''}} have [[countably many]] components, and the rest are connected.