Editing Orthogonal group (section)

== Topology ==
{{confusing section|reason=most notations are undefined; no context for explaining why these consideration belong to the article. Moreover, the section consists essentially in a list of advanced results without providing the information that is needed for a non-specialist for verifying them (no reference, no link to articles about the methods of computation that are used, no sketch of proofs|date=November 2019}}
{{technical |section|date=November 2019}}

=== Low-dimensional topology ===

The low-dimensional (real) orthogonal groups are familiar [[topological space|space]]s:
* {{math|1=O(1) = ''S''<sup>0</sup>}}, a [[2 (number)|two]]-point [[discrete topology|discrete space]]
* {{math|1=SO(1) = {1} }}
* {{math|SO(2)}} is {{math|[[circle|''S''<sup>1</sup>]]}}
* {{math|[[Rotation group SO(3)|SO(3)]]}} is {{math|[[real projective space|'''R'''P<sup>3</sup>]]}} <ref>{{harvnb|Hall|2015}} Section 1.3.4</ref>
* {{math|SO(4)}} is [[double cover (topology)|doubly covered]] by {{math|1=[[special unitary group|SU]](2) × SU(2) = [[3-sphere|''S''<sup>3</sup>]] × ''S''<sup>3</sup>}}.

=== Fundamental group ===
In terms of [[algebraic topology]], for {{math|''n'' > 2}} the [[fundamental group]] of {{math|SO(''n'', '''R''')}} is [[Cyclic group|cyclic of order 2]],<ref>{{harvnb|Hall|2015}} Proposition 13.10</ref> and the [[spin group]] {{math|Spin(''n'')}} is its [[universal cover]]. For {{math|1=''n'' = 2}} the fundamental group is [[infinite cyclic]] and the universal cover corresponds to the  [[real line]] (the group {{math|Spin(2)}} is the unique connected [[double cover (topology)|2-fold cover]]).

=== Homotopy groups ===
Generally, the [[homotopy group]]s {{math|π<sub>''k''</sub>(''O'')}} of the real orthogonal group are related to [[homotopy groups of spheres]], and thus are in general hard to compute. However, one can compute the homotopy groups of the stable orthogonal group (aka the infinite orthogonal group), defined as the [[direct limit]] of the sequence of inclusions:
: <math>\operatorname{O}(0) \subset \operatorname{O}(1)\subset \operatorname{O}(2) \subset \cdots \subset O = \bigcup_{k=0}^\infty \operatorname{O}(k)</math>

Since the inclusions are all closed, hence [[cofibration]]s, this can also be interpreted as a union. On the other hand, {{math|[[n-sphere|''S''<sup>''n''</sup>]]}} is a [[homogeneous space]] for {{math|O(''n'' + 1)}}, and one has the following [[fiber bundle]]:
: <math>\operatorname{O}(n) \to \operatorname{O}(n + 1) \to S^n,</math>
which can be understood as "The orthogonal group {{math|O(''n'' + 1)}} acts [[transitive action|transitively]] on the unit sphere {{math|''S''<sup>''n''</sup>}}, and the [[stabilizer (group theory)|stabilizer]] of a point (thought of as a [[unit vector]]) is the orthogonal group of the [[orthogonal complement|perpendicular complement]], which is an orthogonal group one dimension lower." Thus the natural inclusion {{math|O(''n'') → O(''n'' + 1)}} is [[n-connected|{{math|(''n'' − 1)}}-connected]], so the homotopy groups stabilize, and {{math|1=π<sub>''k''</sub>(O(''n'' + 1)) = π<sub>''k''</sub>(O(''n''))}} for {{math|''n'' > ''k'' + 1}}: thus the homotopy groups of the stable space equal the lower homotopy groups of the unstable spaces.

From [[Bott periodicity]] we obtain {{math|Ω<sup>8</sup>''O''  ≃ ''O''}}, therefore the homotopy groups of {{math|''O''}} are 8-fold periodic, meaning {{math|1=π<sub>''k'' + 8</sub>(''O'') = π<sub>''k''</sub>(''O'')}}, and so one need list only the first 8 homotopy groups:
: <math>\begin{align}
  \pi_0 (O) &= \mathbf{Z} / 2\mathbf{Z}\\
  \pi_1 (O) &= \mathbf{Z} / 2\mathbf{Z}\\
  \pi_2 (O) &= 0\\
  \pi_3 (O) &= \mathbf{Z}\\
  \pi_4 (O) &= 0\\
  \pi_5 (O) &= 0\\
  \pi_6 (O) &= 0\\
  \pi_7 (O) &= \mathbf{Z}
\end{align}</math>

==== Relation to KO-theory ====
Via the [[clutching construction]], homotopy groups of the stable space {{math|''O''}} are identified with stable vector bundles on spheres ([[up to isomorphism]]), with a dimension shift of 1: {{math|1=π<sub>''k''</sub>(''O'') = π<sub>''k'' + 1</sub>(''BO'')}}. Setting {{math|1=''KO'' = ''BO'' × '''Z''' = Ω<sup>−1</sup>''O'' × '''Z'''}} (to make {{math|π<sub>0</sub>}} fit into the periodicity), one obtains:
: <math>\begin{align}
  \pi_0 (KO) &= \mathbf{Z}\\
  \pi_1 (KO) &= \mathbf{Z} / 2\mathbf{Z}\\
  \pi_2 (KO) &= \mathbf{Z} / 2\mathbf{Z}\\
  \pi_3 (KO) &= 0\\
  \pi_4 (KO) &= \mathbf{Z}\\
  \pi_5 (KO) &= 0\\
  \pi_6 (KO) &= 0\\
  \pi_7 (KO) &= 0
\end{align}</math>

==== 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.

==== 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)}}.

==== Whitehead tower ====
The orthogonal group anchors a [[Whitehead tower]]:
: <math>\cdots \rightarrow \operatorname{Fivebrane}(n) \rightarrow \operatorname{String}(n) \rightarrow \operatorname{Spin}(n) \rightarrow \operatorname{SO}(n) \rightarrow \operatorname{O}(n)</math>
which is obtained by successively removing (killing) homotopy groups of increasing order. This is done by constructing [[short exact sequence]]s starting with an [[Eilenberg&ndash;MacLane space]] for the homotopy group to be removed. The first few entries in the tower are the [[spin group]] and the [[string group]], and are preceded by the [[fivebrane group]].  The homotopy groups that are killed are in turn {{pi}}<sub>0</sub>(''O'') to obtain ''SO'' from ''O'', {{pi}}<sub>1</sub>(''O'') to obtain ''Spin'' from ''SO'', {{pi}}<sub>3</sub>(''O'') to obtain ''String'' from ''Spin'', and then {{pi}}<sub>7</sub>(''O'') and so on to obtain the higher order [[brane]]s.