Editing Orthogonal group (section)

==== Interpretation of homotopy groups ====
In a nutshell:<ref>{{Cite web|title=Week 105 |website=This Week's Finds in Mathematical Physics |first=John |last=Baez |author-link=John C. Baez |url=https://math.ucr.edu/home/baez/week105.html|access-date=2023-02-01}}</ref>
* {{math|1=π<sub>0</sub>(''K''O) = '''Z'''}} is about [[Hamel dimension|dimension]]
* {{math|1=π<sub>1</sub>(''K''O) = '''Z''' / 2'''Z'''}} is about [[orientation (mathematics)|orientation]]
* {{math|1=π<sub>2</sub>(''K''O) = '''Z''' / 2'''Z'''}} is about [[spin group|spin]]
* {{math|1=π<sub>4</sub>(''K''O) = '''Z'''}} is about [[topological quantum field theory]].

Let {{math|''R''}} be any of the four [[division algebra]]s {{math|'''R'''}}, {{math|'''C'''}}, {{math|'''[[quaternions|H]]'''}}, {{math|'''[[octonions|O]]'''}},<!-- since '''H''' and '''O''' are not fields, and also to avoid conflict with the lead, I change the former ''F'' notation to ''R'' (ring) --Incnis Mrsi --> and let {{math|''L<sub>R</sub>''}} be the [[tautological line bundle]] over the [[projective line]] {{math|''R''P<sup>1</sup>}}, and {{math|[''L<sub>R</sub>'']}} its class in K-theory. Noting that {{math|1=[[real projective line|'''R'''P<sup>1</sup>]] = ''S''<sup>1</sup>}}, {{math|1=[[Riemann sphere|'''C'''P<sup>1</sup>]] = ''S''<sup>2</sup>}}, {{math|1='''H'''P<sup>1</sup> = ''S''<sup>4</sup>}}, {{math|1='''O'''P<sup>1</sup> = ''S''<sup>8</sup>}}, these yield vector bundles over the corresponding spheres, and
* {{math|π<sub>1</sub>(''K''O)}} is generated by {{math|[''L''<sub>'''R'''</sub>]}}
* {{math|π<sub>2</sub>(''K''O)}} is generated by {{math|[''L''<sub>'''C'''</sub>]}}
* {{math|π<sub>4</sub>(''K''O)}} is generated by {{math|[''L''<sub>'''H'''</sub>]}}
* {{math|π<sub>8</sub>(''K''O)}} is generated by {{math|[''L''<sub>'''O'''</sub>]}}

From the point of view of [[symplectic geometry]], {{math|1=π<sub>0</sub>(''K''O) ≅ π<sub>8</sub>(''K''O) = '''Z'''}} can be interpreted as the [[Maslov index]], thinking of it as the fundamental group {{math|π<sub>1</sub>(U/O)}} of the stable [[Lagrangian Grassmannian]] as {{math|U/O ≅ Ω<sup>7</sup>(''K''O)}}, so {{math|1=π<sub>1</sub>(U/O) = π<sub>1+7</sub>(''K''O)}}.