quaternion group

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## Idea

The _quaternion group_ of [[order of a group|order]] 8, $Q_8$, is the [[finite subgroup of SU(2)]] $Q_8 \subset SU(2) \simeq S^3 \subset \mathbb{H}$ of unit [[quaternions]] which consists of the canonical four [[linear basis|basis]]-quaternions and their negatives:

$$
  Q_8
  \;=\;
  \big\{
     \pm 1,
     \,
     \pm i,
     \,
     \pm j,
     \,
     \pm k
  \big\}
  \,.
$$

This is [[isomorphic|isomorphic]] to the [[binary dihedral group]] of the same [[order of a group|order]] $Q_8 \simeq 2 D_4$. As such, the [[Dynkin diagram]] that corresponds to $Q_8$ under the [[ADE-classification]] of [[finite subgroups of SU(2)]] is [[D4]], the [[triality]]-invariant one.

> graphics grabbed from Wikipedia [here](https://upload.wikimedia.org/wikipedia/commons/5/59/Dynkin_diagram_D4.png)

This order-8 quaternion group $Q_8$ is the first in a row of generalized quaternion groups, $Q_{2^n}$, which are also examples of  dicyclic groups, which class forms  part of an even larger family. We will treat both general dicyclic groups and the specific example of the quaternion group together.

## Definitions

The _dicyclic_ of order $4n$, $n\geq 2$, is the [[group]] $Dic_n$ defined by the presentation $\langle x,y | x^{2n}= x^{n} y^{-2}=y^{-1}x y x=1\rangle$.

The _quaternion group_ (of order 8) is then $Dic_n$ for $n=2$.

The _generalised quaternion group_ of order $2^{k+1}$ is $Dic_n$ with $n= 2^{k-1}$.

## Properties

### General

* $Dic_n$ is (isomorphic to) a finite subgroup of $\mathbb{H}^\ast$ as can be seen by taking generators $x=j$ and $y=\cos(\pi/n) + i\sin(\pi/n)$. For $n=2$ this simply yields the subgroup generated by $i$ and $j$.

* $Dic_n$ has another presentation as $\langle R, S, T | R^2=S^2=T^n=R S T\rangle$. $R S T$ as a power of each of the generators is central and $Dic_n/\langle R S T\rangle= D_{2n}$, where $D_{2n}=\langle R, S, T | R^2=S^2=T^n=R S T=1\rangle$ is the [[dihedral group]] of order $2n$.

* {#HamiltonianGroup} $Q_8$ is a **Hamiltonian group** i.e. a [[non-abelian group]] such that every [[subgroup]] is [[normal subgroup|normal]]. Moreover, a general structure theorem for Hamiltonian groups by Baer (1933) says that every Hamiltonian group has a [[direct product group]]-decomposition containing $Q_8$ as a factor hence, in particular, every Hamiltonian group contains $Q_8$ as a [[subgroup]]! (cf. [Scott (1987, p.253](#Scott87)))

* $Q_8$ is the multiplicative part of the quaternionic near-field $J_9$. (cf. [Weibel (2007)](#Weibel07))

+-- {: .num_prop #InclusionInLargerFininteSubgroupsOfSU2}
###### Proposition
**(inclusion of $Q_8$ into [[finite subgroups of SU(2)]])**

Among the [[finite subgroups of SU(2)]] (hence among all "finite quaternion groups") the quaternion group of [[order of a group|order]] 8, $Q_8$ is a proper [[subgroup]] precisely of the three [[ADE-classification|exceptional cases]]:

  * $Q_8 \subset 2 T$ the [[binary tetrahedral group]] ([[normal subgroup|normal]]),

  * $Q_8 \subset 2 O$ the [[binary octahedral group]] ([[normal subgroup|normal]])

  * $Q_8 \subset 2 I$ the [[binary icosahedral group]] (not [[normal subgroup|normal]])

=--

(e.g. [Koca-Moc-Koca 16, p. 8](#KocaMocKoca16), pointing to [Coxeter-Moser 65](#CoxeterMoser65) and [Coxeter 73](#Coxeter73))

### Character table

[[!include character table of 2D4=Dic2=Q8]]

### Matrix representation

There are lots of different ways of defining $Q:=Q_8$. One is that it is the subgroup of $Gl(2,\mathbb{C})$ generated by the matrices

$$\xi = \left(\array{i&0\\0&-i}\right)$$

and

$$\eta =\left(\array{0&-1\\1&0}\right).$$

In this form it is a nice exercise to derive a presentation of $Q_8$.  Clearly $\xi^4=1$ and $\eta$ is not in $ \langle \xi\rangle$ as is easiy checked, so the order of this group must be at least 8.

We note that $\eta^2 = \xi^2$ and that $\eta \xi \eta^{-1}= \xi^{-1}$, so a guess for a presentation  would be

$$
\langle x,y : x^4=1, y^2=x^2, y x y^{-1}=x^{-1}\rangle.
$$

Let us call $G$ the group presented by this presentation, then there is an obvious epimorphism from $G$ to $Q$ sending $x$ to $\xi$ and $y$ to $\eta$. This is an isomorphism as will be clear if we show that the order of $G$ is less than of equal to 8.  Now every element of $G$ can be written in the form $x^i y^j$ with $0\leq i\leq 3$ and $0\leq j\leq 1$, since $y x=x^{-1}y$ so powers of $y$ can be shifted to the right in any expression and then if the resulting power of $y$ is greater than 2 we can use $y^2=x^2$ to replace even powers of $y$ by powers of $x$. We must therefore have that the group $G$ must contain at most 8 elements so the above presentation is a presentation of $Q_8$.

### Subgroup lattice
 {#SubgroupLattice}

The following shows the [[subgroup lattices]] of the first few [[generalized quaternion groups]]:

### Group cohomology

The [[group cohomology]] of the generalized quaternion groups: see [Tomoda & Zvengrowski 2008](#TomodaZvengrowski08)

## Related entries

* [[quaternion]]

* [[dihedral group]]

* [[Pauli group]], [[stabilizer code]]

* [[canonical formula of myth]]

## References

* [[Kenneth Brown|Kenneth S. Brown]], _Cohomology of Groups_ , GTM **87** Springer Heidelberg 1982. (pp.98-101)

* H. S. M. Coxeter, _The binary polyhedral groups, and other generalizations of the quaternion group_ , Duke Math. J. **7** no.1 (1940) pp.367--379.

* T. Y. Lam, _Hamilton's Quaternions_ , pp.429-454 in _Handbook of Algebra III_ , Elsevier Amsterdam 2004.  ([preprint](http://math.berkeley.edu/~lam/quat.ps))

* {#Scott87}W. R. Scott, _Group Theory_ , Dover New York 1987. (pp.189-194, 252-254)

* {#Weibel07}[[Charles Weibel]], _Survey of Non-Desarguesian Planes_ , Notices of the AMS **54**  no.10 (2007) pp.1294–1303. ([pdf](http://www.ams.org/notices/200710/tx071001294p.pdf))

* {#KocaMocKoca16} Mehmet Koca, Ramazan Koç, Nazife Ozdes Koca, _Two groups $2^3.PSL_2(7)$ and $2^3:PSL_2(7)$ of order 1344_ ([arXiv:1612.06107](https://arxiv.org/abs/1612.06107))

* {#CoxeterMoser65} H.S.M.  Coxeter, W.  O.  J.  Moser, _Generators  and  Relations  for  Discrete Groups_, (Springer  Verlag,  1965);

* {#Coxeter73} H.S.M.  Coxeter,  _Regular  Complex  Polytopes_ (Cambridge; Cambridge University Press, 1973).

See also

* Wikipedia, _[Quaternion group](https://en.wikipedia.org/wiki/Quaternion_group)_

* [[Groupprops]], _[Linear representation theory of the quaternion group](https://groupprops.subwiki.org/wiki/Linear_representation_theory_of_quaternion_group)_

* [[GroupNames]], _[Quaternion groups $Q_{2^n}$](https://people.maths.bris.ac.uk/~matyd/GroupNames/quaternion.html)_

On the [[group cohomology]]:

* {#TomodaZvengrowski08} [[Satoshi Tomoda]], [[Peter Zvengrowski]], *Remarks on the cohomology of finite fundamental groups of 3-manifolds*, Geom. Topol. Monogr. 14 (2008) 519-556 ([arXiv:0904.1876](https://arxiv.org/abs/0904.1876))

Discussion of [[equivariant ordinary cohomology]] ([[Bredon cohomology]]) over the point but in arbitrary [[RO(G)-degree]] for [[equivariance group]] the [[quaternion group]]:

* [[Yunze Lu]], *On the $RO(G)$-graded coefficients of $Q_8$ equivariant cohomology*, Topology and its Applications, 2021 ([doi:10.1016/j.topol.2021.107921](https://doi.org/10.1016/j.topol.2021.107921))

[[!redirects quaternion groups]]

[[!redirects quaternionic group]]
[[!redirects quaternionic groups]]

[[!redirects generalized quaternionic group]]
[[!redirects generalized quaternionic groups]]

[[!redirects generalized quaternion group]]
[[!redirects generalized quaternion groups]]

[[!redirects Q8]]


quaternion-Kähler manifold

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## Definition

A [[Riemannian manifold]] $(X,g)$ of [[dimension]] $4n$ for $n \geq 2$ is called a _quaternion-Kähler manifold_ if its [[holonomy group]] is a [[subgroup]] of [[Sp(n).Sp(1)]] (where [[Sp(n)]] is the $n$th [[quaternionic unitary group]], and in particular $Sp(1) \simeq SU(2) \simeq $ [[Spin(3)]], and the [[central product]] is the [[quotient group]] of the [[direct product group]] by the [[diagonal]] [[center]] $\mathbb{Z}/2$).

If the [[holonomy]] group is in fact a [[subgroup]] of just the $Sp(n)$-factor, one speaks of a _[[hyperkähler manifold]]_.

Quaternion-Kähler manifolds are necessarily [[Einstein manifolds]] (see [below](#AsEinsteinManifolds)). In particular their [[scalar curvature]] $R$ is [[constant function|constant]], and hence a [[real number]] $R \in \mathbb{R}$.  If the scalar curvature is [[positive number|positive]], then one speaks of a _positive quaternion-Kähler manifold_.

\linebreak

## Properties

### As part of the Berger classification

[[!include special holonomy table]]

### As $\mathbb{H}$-Riemannian manifolds

[[!include normed division algebra Riemannian geometry -- table]]

### As quaternionic manifolds

+-- {: .num_example #QuaternionKaehlermanifoldsAreQuaternionicManifolds}
###### Example
**([[quaternion-Kähler manifolds]] are [[quaternionic manifolds]])**

By definition, a [[quaternion-Kähler manifold]] $M$ has [[holonomy group]] contained in the [[direct product group]] [[Sp(n)]]$\times$[[Spin(3)|Sp(1)]], admitting an extension of the [[Levi-Civita connection]] $\nabla$ on the holonomy bundle as [[torsion of a G-structure|torsion]]-free. Thus a quaternion-Kähler manifold is automatically a [[quaternionic manifold]].

Such extension $\nabla_\text{quat}$ of $\nabla$ however is not unique, since $\nabla_\text{quat} + \mathcal{S}$ is another Sp(n)Sp(1)-preserving connection, where $\mathcal{S}$ is a (1, 2)-tensor such that for every $p \in M$, $\mathcal{S}(p)$ takes values in the first [[prolongation]] of the [[Lie algebra]] for the [[G-structure]].

=--

### As Einstein manifolds
 {#AsEinsteinManifolds}

[[quaternion-Kähler manifolds]] are [[Einstein manifolds]] (e.g. [Cortés 05, slide 22](#Cortes05))

### Characteristic classes

+-- {: .num_prop #CharacteristicClassesForSpin5Spin3Structure}
###### Proposition

Let $X$ be a  [[closed manifold|closed]] [[smooth manifold]] of [[dimension]] 8 with [[Spin structure]]. If the [[frame bundle]] moreover admits [[G-structure]] for

$\;\;G = $ [[Sp(n).Sp(1)|Sp(2).Sp(1)]] $\hookrightarrow$ [[SO(8)]]

then the [[Euler class]] $\chi$, the [[second Pontryagin class]] $p_2$ and the [[cup product]]-square $(p_1)^2$ of the [[first Pontryagin class]] of the [[frame bundle]]/[[tangent bundle]] are related by

\[
  \label{EulerClassInTermsOfPontryagin}
  8 \chi
  \;=\;
  4 p_2 - (p_1)^2
  \,.
\]

=--

([Čadek-Vanžura 98, Theorem 8.1 with Remark 8.2](#CadekVanzura98))

+-- {: .num_remark}
###### Remark

The same conclusion (eq:EulerClassInTermsOfPontryagin) also holds for $Spin(7)$-structure, see [there](Spin7-manifold#CharacteristicClassesForSpinStructure)

=--

See also at _[[C-field tadpole cancellation]]_.

### Reduction to hyper-Kähler structure

A quaternion-Kähler manifold $(X,g)$ is a [[hyper-Kähler manifold]], hence has $Sp(n) \hookrightarrow Sp(n)\cdot Sp(1)$-structure, precisely if its [[scalar curvature]], which is a [[constant function|constant]] by $(X,g)$ being an [[Einstein manifold]], vanishes: $R(g) = 0$.

(e.g. [Amann 09, below Def. 1.5](#Amann09))

### Positive quaternion-Kähler manifolds
 {#PositiveQuaternionKaehlerManifolds}

+-- {: .num_defn #PositiveQuaternionKaehlerManifold}
###### Definition

A quaternion-Kähler manifold $(X,g)$ is called _positive_ if

1. it is a [[geodesically complete]]

1. its [[scalar curvature]], which is a [[constant function|constant]] by $(X,g)$ being an [[Einstein manifold]], is a [[positive number]], $R(g) \gt 0$.

=--

([Salamon 82, Section 6](#Salamon82), see e.g. [Amann 09, Def. 1.5](#Amann09))

+-- {: .num_prop}
###### Proposition

A [[connected topological space|connected]] [[positive quaternion-Kähler manifold]] (Def. \ref{PositiveQuaternionKaehlerManifold}) is necessarily [[compact topological space|compact]].

=--

([Salamon 82, p. 158 (16 of 29)](#Salamon82))

+-- {: .num_prop }
###### Proposition

A [[connected topological space|connected]] [[positive quaternion-Kähler manifold]] (Def. \ref{PositiveQuaternionKaehlerManifold}) is necessarily [[simply connected topological space|simply connected]].

=--

([Salamon 82, Theorem 6.6](#Salamon82))

+-- {: .num_prop}
###### Proposition

For each [[dimension]] $dim(X)$ there is a [[finite number]] of [[isometry]] [[isomorphism class|classes]] of [[positive quaternion-Kähler manifolds]] (Def. \ref{PositiveQuaternionKaehlerManifold}).

=--

([LeBrun-Salamon 94, Theorem 0.1](#LeBrunSalamon94))

+-- {: .num_prop #WoldSpacesArePositiveQuaternionKaehler}
###### Proposition
(**[[Wolf spaces]] are [[positive quaternion-Kähler manifolds]])**

Every [[Wolf space]] is a [[positive quaternion-Kähler manifold]].

=--

In fact the [[Wolf spaces]] are the only known examples  of [[positive quaternion-Kähler manifold]] (which is not hyper-Kähler ?!), as of today (e.g. [Salamon 82, Section 5](#Salamon82)).

This leads to the **conjecture** that in every dimension, the [[Wolf spaces]] are the only [[positive quaternion-Kähler manifolds]].

The conjecture has been proven for the following [[dimensions]]

* $d = 4$ (Hitchin)

* $d = 8 $ ([Poon-Salamon 91](#PoonSalamon91), [LeBrun-Salamon 94](#LeBrunSalamon94))

\linebreak

## Examples

The archetypical example is

* [[quaternionic projective plane]] $\,\mathbb{H}P^2$

This is the first of the list of examples  of spaces that are both [[quaternion-Kähler manifolds]]  as well as [[symmetric spaces]], called _[[Wolf spaces]]_.

See around Prop. \ref{WoldSpacesArePositiveQuaternionKaehler} above.

\linebreak

## Related concepts

* [[quaternionic manifold]], [[Kähler manifold]], [[hyper-Kähler manifold]]

## References

Original articles:

* {#Salamon82} [[Simon Salamon]], _Quaternionic Kähler manifolds_, Invent Math (1982) 67: 143. ([doi:10.1007/BF01393378](https://doi.org/10.1007/BF01393378))

* {#PoonSalamon91} Y. S. Poon, [[Simon Salamon]], _Quaternionic Kähler 8-manifolds with positive scalar curvature_, J. Differential Geom. Volume 33, Number 2 (1991), 363-378 ([euclid:1214446322](https://projecteuclid.org/euclid.jdg/1214446322))

* {#LeBrunSalamon94} Claude LeBrun, [[Simon Salamon]], _Strong rigidity of positive quaternion Kähler manifolds_, Inventiones Mathematicae 118, 1994, 109–132 ([dml:144231](https://eudml.org/doc/144231), [doi:10.1007/BF01231528](https://doi.org/10.1007/BF01231528))

Exposition:

* {#Cortes05} [[Vicente Cortés]], _Quaternionic Kähler manifolds_, 2005 ([pdf](https://www2.math.hu-berlin.de/gradkoll/Cortes_vorlesung1_handout.pdf))

Textbook accounts:

* {#Besse}[[Arthur Besse]], _Einstein Manifolds_, Springer-Verlag 1987.

* {#Joyce} [[Dominic Joyce]], _Compact Manifolds with Special Holonomy_, Oxford University Press, 2000.

See also

* Wikipedia, _[Quaternion-Kähler manifold](https://en.wikipedia.org/wiki/Quaternion-Kaehler_manifold)_

In terms of [[G-structure]]:

* Edmond Bonan, _Sur les $G$-structures de type quaternionien_,  Cahiers de Topologie et Géométrie Différentielle Catégoriques, Volume 9 (1967) no. 4, p. 389-463 ([numdam:CTGDC_1967__9_4_389_0](http://www.numdam.org/item/CTGDC_1967__9_4_389_0))

* {#Salamon86} [[Simon Salamon]], _Differential Geometry of Quaternionic Manifolds_, Annales scientifiques de l’É.N.S. 4e série, tome 19, no 1 (1986), p. 31-55 ([numdam:ASENS_1986_4_19_1_31_0](http://www.numdam.org/item/ASENS_1986_4_19_1_31_0/))

* [[Dmitri V. Alekseevsky]], [[Stefano Marchiafava]], _Quaternionic-like structures on a manifold: Note I. 1-integrability and integrability conditions_, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1993) Volume: 4, Issue: 1, page 43-52 ([dml:244082](https://eudml.org/doc/244082))

* [[Dmitri V. Alekseevsky]], [[Stefano Marchiafava]], _Quaternionic-like structures on a manifold: Note II. Automorphism groups and their interrelations_, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1993) Volume: 4, Issue: 1, page 53-61 ([dml:244299](https://eudml.org/doc/244299))

* [[Dmitry V. Alekseevsky]], [[Stefano Marchiafava]], _Quaternionic structures on a manifold and subordinated structures_, Annali di Matematica pura ed applicata 171, 205–273 (1996) ([doi:10.1007/BF01759388](https://doi.org/10.1007/BF01759388))

* Joseph Thurman, _Quaternionic Geometry and Special Holonomy_, 2018 ([pdf](http://www.math.stonybrook.edu/alumni/2018-Joseph-Thurman.pdf))

Articles discussing quaternion-Kähler [[holonomy]], [[connection]], and relation to other [[hypercomplex manifold|hypercomplex structures]]:

* {#Moroianu} [[Andrei Moroianu]], [[Uwe Semmelmann]], _Killing Forms on Quaternion-Kähler Manifolds_, Annals of Global Analysis and Geometry, November 2005, Volume 28, Issue 4, pp 319–335 ([arXiv:math/0403242](https://arxiv.org/abs/math/0403242), [doi:10.1007/s10455-005-1147-y](https://doi.org/10.1007/s10455-005-1147-y))

* {#PerdersonSwannPoon} Pedersen, Poon, and Swann. "Hypercomplex structures associated to quaternionic manifolds",  Differential Geometry and its Applications (1998) 273-293 North-Holland.

* {#Verbitsky} [[Misha Verbitsky]], _Hyperkähler manifolds with torsion, supersymmetry and Hodge theory_, Asian J. Math, V. 6 No. 4, pp. 679-712, Dec. 2002.

See also

* Claude LeBrun, _On complete quaternionic-Kähler manifolds_, Duke Math. J. Volume 63, Number 3 (1991), 723-743 ([euclid:1077296077](https://projecteuclid.org/euclid.dmj/1077296077))

* Simon G. Chiossi, Óscar Maciá, _SO(3)-Structures on 8-manifolds_, Ann. Glob. Anal. Geom. 43 (1) (2013), 1--18 ([arXiv:1105.1746](https://arxiv.org/abs/1105.1746))

On [[positive quaternion-Kähler manifolds]]

* {#Amann09} Amann, _Positive Quaternion Kähler Manifolds_, 2009 ([pdf](https://d-nb.info/996176438/34))

* Amann, _Partial Classification Results for Positive Quaternion Kaehler Manifolds_ ([arXiv:0911.4587](https://arxiv.org/abs/0911.4587))

Discussion of [[characteristic classes]]:

* {#CadekVanzura98} [[Martin Čadek]], [[Jiří Vanžura]], _Almost quaternionic structures on eight-manifolds_, Osaka J. Math. Volume 35, Number 1 (1998), 165-190 ([euclid:1200787905](https://projecteuclid.org/euclid.ojm/1200787905))

On [[quaternion-Kähler orbifolds]]:

*  K. Galicki, [[H. Blaine Lawson]], _Quaternionic Reduction and Quaternionic Orbifolds_, Mathematische Annalen (1988) Volume: 282, Issue: 1, page 1-22 ([dml:164446](https://eudml.org/doc/164446))

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quaternionic Hopf fibration

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## Idea

The _quaternionic Hopf fibration_ is the [[Hopf fibration]] induced by the [[quaternions]], hence it is the [[fibration]]

$$
  \array{
     S^3 &\hookrightarrow& S^7
     \\
     &&
     \Big\downarrow^{\mathrlap{p_{\mathbb{H}}}}
     \\
     && S^4
  }
$$

of the [[7-sphere]] over the [[4-sphere]] with [[fiber]] the [[3-sphere]], which is induced via the [[Hopf construction]] from the product operation

$$
  \mathbb{H} \times \mathbb{H} \stackrel{(-)\cdot (-)}{\longrightarrow}
  \mathbb{H}
$$

on the [[quaternions]], or else from

$$
  \mathbb{H} \times \mathbb{H}^{\times} \stackrel{(-)\cdot (-)^{-1}}{\longrightarrow}
  \mathbb{H}
$$

to match standard conventions.

This means that if $S^7$ is regarded as the [[unit sphere]] $\{(x,y)  | {\vert x\vert}^2 + {\vert y\vert}^2 = 1\}$ in $\mathbb{H}\times \mathbb{H}$ and $S^4$ is regarded as the  [[quaternionic projective space]], then $p$ is given (on points $(x,y)$ with $y \neq 0$) simply by

$$
  p_{\mathbb{H}} \colon (x,y) \mapsto [x;y] = [x/y; 1]
  \,,
$$

## Properties

### $SO(3)$- and $Spin(5)$-Equivariant structure
 {#EquivariantStructure}

Since the [[automorphism group]] of the [[quaternions]], as an $\mathbb{R}$-[[associative algebra|algebra]], is the [[special orthogonal group]] $SO(3)$

$$
  \mathrm{Aut}_{\mathbb{R}}(\mathbb{H}) \simeq SO(3)
$$

acting by [[rotation]] of the imaginary quaternions, via the [[Hopf construction]] it follows that the 7-sphere and 4-sphere inherit $SO(3)$-[[actions]] under which the quaternionic Hopf map is equivariant.

Notice that this means that $SO(3)$ acts on $S^7$ here diagonally on the _two_ copies of the imaginary octonions in $S^7 \hookrightarrow \mathbb{H} \oplus \mathbb{H}$ (as opposed to, say, via any one of the embeddings $SO(3) \hookrightarrow SO(8)$ and the following canonical action of $SO(8)$ on $S^7 \hookrightarrow \mathbb{R}^8$).

(see also [Cook-Crabb 93](#CookCrabb93))

But in fact more is true:

+-- {: .num_prop #Spin5EquivarianceOfQuaternionicHopfFibration}
###### Proposition
**([[Spin(5)]]-[[action|equivariance]] of [[quaternionic Hopf fibration]])**

Consider

1. the [[Spin(5)]]-[[action]] on the [[4-sphere]] $S^4$ which is induced by the defining action on $\mathbb{R}^5$ under the identification $S^4 \simeq S(\mathbb{R}^5)$;

1. the [[Spin(5)]]-action on the [[7-sphere]] $S^7$ which is induced under the exceptional [[isomorphism]] $Spin(5) \simeq Sp(2) = U(2,\mathbb{H})$ (this Prop. \ref{Spin(5)#ExceptionalIsoToSp2}) by the canonical left action of $U(2,\mathbb{H})$ on $\mathbb{H}^2$ via $S^7 \simeq S(\mathbb{H}^2)$.

Then the [[quaternionic Hopf fibration]] $S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4$ is [[equivariant]] with respect to these [[actions]].

=--

This appears as ([Gluck, Warner & Ziller (1986), Prop. 4.1](#GluckWarnerZiller86)).

The statement is also almost explicit in [Porteous 95, p. 263](#Porteous95)

\begin{center}
\begin{imagefromfile}
  "file_name": "Spin5OnHopfH.jpg",
  "width": 280
\end{imagefromfile}
\end{center}

A way to make the $Spin(5)$-equivariance of the quaternionic Hopf fibration fully explicit is to observe that the quaternionic Hopf fibration is equivalently the following map of [[coset spaces]]:

$$
  \array{
    S^3
      &\overset{fib(h_{\mathbb{H}})}{\longrightarrow}&
    S^{7}
      &\overset{h_{\mathbb{H}}}{\longrightarrow}&
    S^4
    \\
     = && = && =
    \\
    \frac{Spin(4)}{Spin(3)}
    &\longrightarrow&
    \frac{Spin(5)}{Spin(3)}
    &\longrightarrow&
    \frac{Spin(5)}{Spin(4)}
  }
$$

([Hasuda-Tomizawa 09, table 1](#HasudaTomizawa09))

+-- {: .num_remark}
###### Remark

Of the resulting [[action]] of [[Sp(2)]]$\times$[[Sp(1)]] on the [[7-sphere]] (from Prop. \ref{Spin5EquivarianceOfQuaternionicHopfFibration}), only the [[quotient group]] [[Sp(2).Sp(1)]] acts [[effective action|effectively]].

=--

### Class in the homotopy groups of spheres
 {#ClassInTheHomotopyGroupsOfSpheres}

The quaternionic Hopf fibration gives an element in the 7th [[homotopy groups of spheres|homotopy group of the 4-sphere]]

$$
  [p_{\mathbb{H}}] \in \pi_7(S^4) \simeq \mathbb{Z} \times (\mathbb{Z}/12)
$$

and in fact it is a generator of the non-torsion factor in this group.

Stably, i.e. as a generator for the [[stable homotopy groups of spheres]] in degree $7-4 = 3$, the quaternionic Hopf map becomes a [[torsion subgroup|torsion]] generator:

$$
  [p_{\mathbb{H}}] \in \pi_3^S \simeq \mathbb{Z}/24
  \,,.
$$

The [[third stable homotopy group of spheres]] (the third [[stable stem]]) is the [[cyclic group]] of [[order of a group|order]] 24:

$$
  \array{
    \pi_3^s &\simeq& \mathbb{Z}/24
    \\
    [h_{\mathbb{H}}] &\leftrightarrow& [1]
  }
$$

where the generator $[1] \in \mathbb{Z}/24$ is represented by the [[quaternionic Hopf fibration]] $S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4$.

Under the [[Pontrjagin-Thom isomorphism]], identifying the [[stable homotopy groups of spheres]] with the [[bordism ring]] $\Omega^{fr}_\bullet$ of [[stable framing|stably framed]] manifolds (see at _[[MFr]]_), this generator is represented by the [[3-sphere]] (with its left-invariant framing induced from the identification with the [[Lie group]] [[SU(2)]] $\simeq$ [[Sp(1)]] )

$$
  \array{
    \pi_3^s & \simeq & \Omega_3^{fr}
    \\
    [h_{\mathbb{H}}] & \leftrightarrow & [S^3]
    \,.
  }
$$

Moreover, the relation $2 4 [S^3] \,\simeq\, 0$ is represented by the [[bordism]] which is the [[complement]] of 24 [[open balls]] inside [[generalized the|the]] [[K3]]-manifold ([Wang-Xu 10, Sec. 2.6](third+stable+homotopy+group+of+spheres#WangXu10)).

### Class in equivariant stable homotopy theory
 {#ClassInEquivariantStableHomotopyTheory}

Fix a [[finite group|finite]] [[subgroup]] $G \hookrightarrow SO(3)$ which does not come from $SO(2) \hookrightarrow SO(3)$ -- i.e. not a [[cyclic group]], but one of the [[dihedral groups]] or else the [[tetrahedral group]] or [[octahedral group]] or [[icosahedral group]] (by the [[ADE classification]]).

Regard both $S^7$ and $S^4$ as pointed [[topological G-spaces]] via the $SO(3)$-action induced via automorphisms of the quaternions, as [above](#EquivariantStructure). Write

$$
  \Sigma^\infty_G S^7, \Sigma^\infty_G S^4 \in G Spectra
$$

for the corresponding [[equivariant suspension spectra]].

Notice that if we took trivial $G$, then in the [[stable homotopy category]]

$$
  [\Sigma^\infty S^7, \Sigma^\infty S^4] \simeq \mathbb{Z}/24
$$

by the [above](#ClassInTheHomotopyGroupsOfSpheres). In contrast:[^acknowledgement]

[^acknowledgement]: The proof of prop. \ref{QuaternionicHopfFibrationIsDEEquivariantlyStablyNonTorsion} profited from [[Charles Rezk]], who suggested [here](http://mathoverflow.net/a/224185/381) that the reduction to fixed points will make the real Hopf fibration give a non-torsion contribution, and from [[David Barnes]] who amplified the use of the Greenless-May splitting theorem.

+-- {: .num_prop #QuaternionicHopfFibrationIsDEEquivariantlyStablyNonTorsion}
###### Proposition

In $G$-[[equivariant homotopy theory]] this becomes a non-[[torsion subgroup|torsion group]], i.e.

$$
  [\Sigma^\infty_G S^7, \Sigma^\infty_G S^4]_G \simeq \mathbb{Z} \oplus \cdots
$$

with the quaternionic Hopf fibration, regarded as a $G$-equivariant map, representing a non-torsion element.

=--

+-- {: .proof}
###### Proof

First use the [Greenlees-May decomposition](rational+equivariant+stable+homotopy+theory#SplittingIntoMackeyFunctors) which says that for any two $G$-[[equivariant spectra]] $X,Y$ and writing $\pi_\bullet(X), \pi_\bullet(Y)$ for their [[equivariant homotopy groups]], organized as [[Mackey functors]] $H \mapsto \pi_n^H(X)$ for all subgroups $H \subset G$, then the canonical map

$$
  [X,Y]_G
  \longrightarrow
   \underset{n}{\oplus}
  Hom_{\mathcal{M}[G]}(\pi_n(X), \pi_n(Y))
$$

is [[rational equivariant stable homotopy theory|rationally]] an [[isomorphism]].

With this we are reduced to showing that there exists $n \in \mathbb{Z}$ and a morphism of [[Mackey functors]] of [[equivariant homotopy groups]] $\pi_n(\Sigma^\infty_G S^7) \to \pi_n(\Sigma^\infty_G S^4)$ which is not a torsion element in the abelian [[hom-object|hom-group]] of Mackey functors.

To analyse this, we use the [[tom Dieck splitting]] which says that the [[equivariant homotopy groups]] of [[equivariant suspension spectra]] $\Sigma^\infty_G X$ contain a [[direct sum|direct summand]] which is simply the ordinary stable homotopy groups of the naive [[fixed point]] space $X^H$:

$$
  \pi_n^H(\Sigma^\infty_G X) \simeq \pi_n(\Sigma^\infty (X^H)) \oplus \cdots
  \,.
$$

Now observe that the [[fixed points]] of the $SO(3)$-action on the quaternionic Hopf fibration that we are considering is just the [[real Hopf fibration]]:

$$
  (p_{\mathbb{H}})^{SO(3)} = p_{\mathbb{R}} \;\colon\; S^1 \longrightarrow S^1
$$

since $SO(3)$ acts transitively on the quaternionic quaternions and fixes the real quaternions.  By our assumption that $G \subset SO(3)$ does not come through $SO(2) \hookrightarrow SO(3)$ it follows that this statment is still true for $G$:

$$
  (p_{\mathbb{H}})^{G} = p_{\mathbb{R}} \;\colon\; S^1 \longrightarrow S^1
  \,.
$$

But the [[real Hopf fibration]] defines a non-torsion element in $\pi_0^S \simeq \mathbb{Z}$.

In conclusion then, at $n = 1$ and $H = G$ we find that the $G$-equivariant quaternionic Hopf fibration contributes a non-torsion element in

$$
  Hom_{Ab}(\pi_1^G(\Sigma^\infty_G S^7), \pi_1^G(\Sigma^\infty_G S^4))
$$

which appears as a non-torsion element in

$$
  Hom_{\mathcal{M}[G]}( \pi_1(\Sigma^\infty_G S^7), \pi_1(\Sigma^\infty_G S^4) )
$$

and hence in $[\Sigma^\infty_G S^7, \Sigma^\infty_G S^4]_G$.

=--

See also at _[[equivariant stable cohomotopy]]_

## Related concepts

* [[real Hopf fibration]], [[complex Hopf fibration]], [[octonionic Hopf fibration]]

* [[Calabi-Penrose fibration]]

* [[quaternionic projective space]]

## References

The original article:

* {#Hopf35} [[Heinz Hopf]], *Über die Abbildungen von Sphären auf Sphäre niedrigerer Dimension*, Fundamenta Mathematicae **25** 1 (1935) 427-440 [[eudml:212801](https://eudml.org/doc/212801)]

Further discussion:

* {#GluckWarnerZiller86} [[Herman Gluck]], [[Frank Warner]], [[Wolfgang Ziller]], _The geometry of the Hopf fibrations_, L'Enseignement Mathématique, **32** (1986), 173-198 [[ResearchGate](https://www.researchgate.net/publication/266548925_The_geometry_of_the_Hopf_fibrations), [[GluckWarnerZiller-HopfFibrations.pdf:file]]]

* {#Miyaoka93} [[Reiko Miyaoka]], _The linear isotropy group of $G_2/SO(4)$, the Hopf fibering and isoparametric hypersurfaces_, Osaka J. Math. **30** 2 (1993) 179-202 [[Euclid](http://projecteuclid.org/euclid.ojm/1200784357)]

* {#Porteous95} [[Ian Porteous]], _Clifford Algebras and the Classical Groups_, Cambridge Studies in Advanced Mathematics, Cambridge University Press (1995) ([doi:10.1017/CBO9780511470912](https://doi.org/10.1017/CBO9780511470912))

* {#HasudaTomizawa09} Machiko Hatsuda, Shinya Tomizawa, *Coset for Hopf fibration and Squashing*, Class. Quant. Grav. **26** 225007 (2009) [[arXiv:0906.1025](https://arxiv.org/abs/0906.1025), [doi:10.1088/0264-9381/26/22/225007](https://doi.org/10.1088/0264-9381/26/22/225007)]

* [[Rustam Sadykov]], §8.3 in: _Elements of Surgery Theory_, 2013 ([pdf](https://www.math.ksu.edu/~sadykov/Lecture%20Notes/Surgery%20Theory.pdf), [[SadykovSurgeryTheory.pdf:file]])

* [[Tyrone Cutler]], p. 23 of: *Fibrations IV* (2020) [[pdf](https://www.math.uni-bielefeld.de/~tcutler/pdf/Fibrations%20IV.pdf), [[Cutler-FibrationsIV.pdf:file]]]

Discussion in [[parameterized homotopy theory]] includes

* {#CookCrabb93} A. L. Cook, M. C. Crabb, _Fiberwise Hopf structures on sphere bundles_, J. London Math. Soc. (2) 48 (1993) 365-384 ([pdf](http://www.maths.ed.ac.uk/~aar/papers/crabbcook.pdf))

* {#Iriye95} [[Kouyemon Iriye]], _Equivariant Hopf structures on a sphere_, J. Math. Kyoto Univ. Volume 35, Number 3 (1995), 403-412 ([Euclid](http://projecteuclid.org/euclid.kjm/1250518704))

Discussion in [[homotopy type theory]] is in

* [[Ulrik Buchholtz]], [[Egbert Rijke]], _The Cayley-Dickson Construction in Homotopy Type Theory_ ([arXiv:1610.01134](https://arxiv.org/abs/1610.01134))

Noteworthy [[fiber products]] with the quaternionic Hopf fibration, notably [[exotic 7-spheres]], are discussed in

* Llohann D. Sperança, _Explicit Constructions over the Exotic 8-sphere_ ([pdf](https://www.ime.unicamp.br/~rigas/sigma8EncontroTopol.pdf), [[SperancaExoticSpheres.pdf:file]])


quaternionic K-theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Algebraic topology
+--{: .hide}
[[!include algebraic topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

What is called *symplectic K-theory* (denoted $K Sp$) or, equivalently, *quaternionic K-theory* (denoted $K Q$) is
the [[topological K-theory]] of [[quaternionic vector bundles]].

This is in direct analogy with how [[complex K-theory]] ([[KU]]) is the topological K-theory of [[complex vector bundles]] and [[KO]] that of [[real vector bundles]].

In fact, quaternionic K-theory is equivalently [[KO-theory]] in degrees which are multiples of 4.

(...)

## Applications

### To topological phases of matter

In [[solid state physics]], under the [[K-theory classification of topological phases of matter]], quaternionic K-theory is thought to classify [[crystal|crystalline]] [[topological insulator]]-phases of materials whose [[electron]]-dynamics respects [[time-reversal symmetry]] (this makes the classification be by [[KO-theory|$KO^\bullet$]]) but *no* [[parity symmetry]] (this restricts the classification further to $KO^{4 n} \simeq K Q^n$). Then spin-less such phases are classified by $K Q^0 \simeq K O^0$, while spin-ful such phases are classified by $K Q^1$.

In this context, the [[Kane-Mele invariant]] is a projection $K Q^1(\mathbb{T}^d) \to \mathbb{Z}/2$ from the quaternionic K-theory of the [[Brillouin torus]] of a material, which detects (in particular, for $d = 2$) the non-triviality of the [[graphene]]-phase (which is a [[time-reversal symmetry|time-reversal]]-symmetric [[topological insulator]] when its [[spin-orbit coupling]] is taken into account).

## Related concepts

* [[KR-theory]], [[KU-theory]], [[KO-theory]]

* [[quaternionic oriented cohomology theory]]

[[!redirects KQ-theory]]


quaternionic Kähler manifold > history

see _[[quaternion-Kähler manifold]]_


quaternionic manifold

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

##Definition

There is some variation in the literature on what one calls a "[[quaternionic]] [[manifold]]''. The most general definition, however, encompassing all the others, is:

+-- {: .num_defn #QuaternionicManifold}
###### Definition
**(Quaternionic manifold)**

For $n \geq 2$, a **quaternionic manifold** is a real 4n-dimensional manifold M with a $\text{GL} (n,\mathbf{H})\cdot \mathbf{H} ^{\times }$-[[G-structure|structure]] which admits a [[torsion of a G-structure|torsion-free]] [[connection]] $\nabla$ (i.e. is integrable as a [[G-structure]]).

=--

When $\nabla$ does not exist $M$ is called _almost quaternionic_, and this is just a [[reduction of the structure group]] of $M$ along a [[Lie group]] inclusion $\text{GL}(n, \mathbf{H}) \hookrightarrow \text{GL}(4n, \mathbf{R})$. Note that the multiplicative group of the quaternions $\mathbf{H}^{\times}$ can be normalized, so that the second factor of $\text{GL} (n,\mathbf{H})\cdot \mathbf{H} ^{\times }$ is isomorphic to SU(2), or isomorphically again the first quaternionic unitary group Sp(1). Hence one occasionally finds the structure group reduction written $\text{GL} (n,\mathbf{H})\cdot \text{Sp}(1)$.

+-- {: .num_defn}
###### Remark

Other definitions have been given, based on the following useful but more naïve reasoning: consider two  [[almost complex structures]] on a smooth $4n$-manifold $M$, say $\langle M, I \rangle$ and $\langle M, J \rangle$, given the relation $\{ I, J \} =0$. Then call the structure $\langle M, I, J \rangle$ "almost quaternionic", and a map $\varphi:  \langle M, I, J \rangle \rightarrow \langle N, K, F \rangle$ of almost quaternionic structures "quaternionic" if it is separately [[complex analytic]] as a map $\langle M, I \rangle \rightarrow \langle N, K \rangle$ and also as a map $\langle M, J \rangle \rightarrow \langle N, F \rangle$.

When $\mathbf{R}^4$ is given a quaternionic multiplication with anti-commuting imaginary units $i$ and $j$, this is actually equivalent to $\varphi : \mathbf{R}^4 \rightarrow \mathbf{R}^4$ satisfying the [[Cauchy-Feuter]] complex:
$$ i \frac{\partial}{\partial u_3} = \frac{\partial }{\partial u _4}, j \frac{\partial }{ \partial u_2} = \frac{\partial }{\partial u_4}$$
$$i \frac{\partial}{\partial u_1} = \frac{\partial }{\partial u_2}, j \frac{\partial }{\partial u_1 } = \frac{\partial }{\partial u_3}$$
known as a starting point for defining a notion of "quaternionic [[holomorphy]]", since the Cauchy-Feuter complex consists of two separate [[Cauchy-Riemann]] systems in the imaginary units $i, j$. Such a complex exists on $\mathbf{R}^{4n}$ for any $n$ as an extended Cauchy-Fueter complex consisting of the systems above, repeated for each set of four coordinates. Hence, $\mathbf{R}^{4n}$ as a smooth manifold can always be endowed with an almost quaternionic structure. On this account of quaternionic structure, a _quaternionic $n$-manifold_ is then an almost quaternionic structure on a smooth manifold $M$ such that $M$ has an atlas of quaternionic maps, considered with respect to the standard quaternionic structure on $\mathbf{R}^{4n}$ just described. However, such a definition has serious drawbacks, such as the fact that quaternionic projective $n$-space $\mathbf{H}P^n := \text{Sp}(n + 1)/\text{Sp}(n)\text{Sp}(1)$ is not a "quaternionic manifold" in this sense for any $n$. It is, however, a quaternionic manifold under the official definition above.

=--

## Examples

+-- {: .num_example #QuaternionKaehlermanifoldsAreQuaternionicManifolds}
###### Example
**([[quaternion-Kähler manifolds]] are [[quaternionic manifolds]])**

By definition, a [[quaternion-Kähler manifold]] $M$ has [[holonomy group]] contained in the [[direct product group]] [[Sp(n)]]$\times$[[Spin(3)|Sp(1)]], admitting an extension of the [[Levi-Civita connection]] $\nabla$ on the holonomy bundle as torsion-free. Thus a quaternion-Kähler manifold is automatically quaternionic.

Such extension $\nabla_\text{quat}$ of $\nabla$ however is not unique, since $\nabla_\text{quat} + \mathcal{S}$ is another Sp(n)Sp(1)-preserving connection, where $\mathcal{S}$ is a (1, 2)-tensor such that for every $p \in M$, $\mathcal{S}(p)$ takes values in the first [[prolongation]] of the [[Lie algebra]] for the [[G-structure]].

=--

## Properties

### Holonomy

(...)

### Twistor Space of a Quaternionic Manifold

(...)

## Related concepts

* [[biquaternionic manifold]], [[hypercomplex manifold]],

* [[quaternionic-Kähler manifold]]

* [[Kähler manifold]], [[hyper-Kähler manifold]]

## References

Book references include:

* {#Besse}[[Arthur Besse]], _Einstein Manifolds_, Springer-Verlag 1987.

* {#Joyce} [[Dominic Joyce]], _Compact Manifolds with Special Holonomy_, Oxford University Press, 2000.

Classical references:

* Edmond Bonan, _Sur les $G$-structures de type quaternionien_,  Cahiers de Topologie et Géométrie Différentielle Catégoriques, Volume 9 (1967) no. 4, p. 389-463 ([numdam:CTGDC_1967__9_4_389_0](http://www.numdam.org/item/CTGDC_1967__9_4_389_0))
* {#Salamon} S.M. Salamon, "Differential Geometry of Quaternionic Manifolds", Annales scientifiques de l’É.N.S. 4e série, tome 19, no 1 (1986), p. 31-55.

See also

* Wikipedia, _[Quaternionic manifold](https://en.wikipedia.org/wiki/Quaternionic_manifold)_

In terms of [[G-structure]]:

* [[Dmitri V. Alekseevsky]], [[Stefano Marchiafava]], _Quaternionic-like structures on a manifold: Note I. 1-integrability and integrability conditions_, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1993) Volume: 4, Issue: 1, page 43-52 ([dml:244082](https://eudml.org/doc/244082))

* [[Dmitri V. Alekseevsky]], [[Stefano Marchiafava]], _Quaternionic-like structures on a manifold: Note II. Automorphism groups and their interrelations_, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1993) Volume: 4, Issue: 1, page 53-61 ([dml:244299](https://eudml.org/doc/244299))

* [[Dmitry V. Alekseevsky]], _Quaternionic structures on a manifold and subordinated structures_, Annali di Matematica pura ed applicata 171, 205–273 (1996) ([doi:10.1007/BF01759388](https://doi.org/10.1007/BF01759388))

Holonomy, connections, and twistor spaces:

* {#ivanov} Stefan Ivanov, Ivan Minchev, Simeon Zamkovoy, [Twistors of Almost Quaternionic Manifolds](https://arxiv.org/abs/math/0511525)

* {#grantscharov} Gueo Grantcharov and Yat Sun Poon, [Geometry of Hyper-Kahler Connections with Torsion](https://arxiv.org/abs/math/9908015v1)

On [[quaternionic orbifolds]]:

*  K. Galicki, [[H. Blaine Lawson]], _Quaternionic Reduction and Quaternionic Orbifolds_, Mathematische Annalen (1988) Volume: 282, Issue: 1, page 1-22 ([dml:164446](https://eudml.org/doc/164446))

[[!redirects quaternionic manifolds]]
[[!redirects quaternion manifold]]

[[!redirects quaternionic orbifold]]
[[!redirects quaternionic orbifolds]]


quaternionic oriented cohomology theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Quaternionic oriented cohomology theory is the analog of [[complex oriented cohomology theory]] as [[complex vector bundles]] are replaced by [[quaternionic vector bundles]]:

## Definition

+-- {: .num_defn #QuaternionicOrientedCohomologyTheory}
###### Definition
**(quaternionic oriented cohomology theory)**

A _quaternionic orientation_ $p_1^E$ (a generalized [[first Pontrjagin class]]) in a [[multiplicative cohomology theory|multiplicative]] [[Whitehead generalized cohomology theory]] $E$ is an [[extension]] of the 4-suspended [[ring unit]] in the [[cohomology ring]] $E_\bullet \,\simeq\, E^4\big( H\mathbb{P}^1\big)$ from the [[quaternionic projective space]] $\mathbb{H}P^1$ (the [[4-sphere]]) to $H\mathbb{P}^\infty$:

\[
  \label{QuaternionicOrientationByExtension}
  \array{
    \mathbb{H}P^1 &\overset{\Sigma^4 1}{\longrightarrow}&
    \Omega^{\infty - 4} E
    \\
    \big\downarrow & \nearrow_{ \mathrlap{p_1^E} }
    \\
    H \mathbb{P}^\infty
  }
\]

=--

+-- {: .num_remark #ComplexOrientationByExtensionsAndTheirObstructions}
###### Remark
**(quaternionic $E$-orientation by [[extensions]] and their [[obstructions]])**

In terms of [[Brown representability theorem|classifying maps]], Def. \ref{QuaternionicOrientedCohomologyTheory} means that a quaternionic orientation $p_1^E$ in $E$-cohomology theory is equivalently an [[extension]] (in the [[classical homotopy category]]) of the map $\Sigma^4 1 \,\colon\, \mathbb{H}P^1 \longrightarrow \Omega^{\infty -4} E$ (which classifies the suspended [[ring unit]] in the [[cohomology ring]]) along the canonical inclusion of [[quaternionic projective spaces]]

\[
  \label{QuaternionicOrientationAsExtension}
  \array{
     \mathbb{H}P^1
     &
     \overset{
       \Sigma^4 1_E
     }{
       \longrightarrow
     }
     &
     \Omega^{\infty - 4} E
     \\
     \big\downarrow
     &
     \nearrow \mathrlap{ {}_{p_1^E} }
     \\
     \mathbb{H}P^\infty
     \,.
  }
\]

Notice that the [[quaternionic projective spaces]] form a [[cotower]]

$$
  \ast
  \,=\,
  \mathbb{H}P^0
  \hookrightarrow
  \mathbb{H}P^1
  \hookrightarrow
  \mathbb{H}P^2
  \hookrightarrow
  \mathbb{H}P^3
  \hookrightarrow
  \cdots
  \hookrightarrow
  \mathbb{H}P^\infty
  \,=\,
  \underset{\longrightarrow}{\lim}
  \mathbb{H}P^\bullet
$$

where each inclusion stage is (by [this Prop.](complex+projective+space#CellComplexStructureOnComplexProjectiveSpace), see at _[[cell structure of projective spaces]]_) the coprojection of a [[pushout]] [of topological spaces](Top#UniversalConstructions) (or rather: of [[pointed topological spaces]]) of the form

$$
  \array{
    D^{4n+4}
    &
    \overset{}{\longrightarrow}
    &
    \mathbb{H}P^{n+1}
    \\
    \big\uparrow
    &\mathclap{^{_{(po)}}}&
    \big\uparrow
    \\
    S^{4n+3}
    &\underset{h^{4n+3}_{\mathbb{H}}}{\longrightarrow}&
    \mathbb{H}P^n
  }
$$

(where $h^{4n+3}_{\mathbb{H}}$ is the [[quaternionic Hopf fibration]] in dimension $4n+3$) hence of a [[homotopy pushout]] of [[classical model structure on topological spaces|underlying homotopy types]] (rather: [[classical model structure on pointed topological spaces|of pointed homotopy types]]) of this form:

$$
  \array{
    \ast
    &
    \overset{}{\longrightarrow}
    &
    \mathbb{H}P^{n+1}
    \\
    \big\uparrow
    &\mathclap{^{_{(hpo)}}}&
    \big\uparrow
    \\
    S^{4n+3}
    &\underset{h^{4n+3}_{\mathbb{H}}}{\longrightarrow}&
    \mathbb{H}P^n
  }
$$

Therefore, a quaternionic orientation by extension (eq:QuaternionicOrientationByExtension) is equivalently the homotopy colimiting map of a sequence

$$
  \big(
    \Sigma^4 1
      \,=\,
    p_1^{E,0}
    ,\,
    p_1^{E,1}
    ,\,
    p_1^{E,2}
    ,\,
    \cdots
  \big)
$$

of finite-stage extensions

$$
  \array{
    \ast
    &
    \overset{}{\longrightarrow}
    &
    \mathbb{H}P^{n+1}
    &
    \overset{ p_1^{E,n+1} }{\longrightarrow}
    &
    \Omega^{\infty - 4} E
    \\
    \big\uparrow
    &\mathclap{^{_{(hpo)}}}&
    \big\uparrow
    &
    \nearrow \mathrlap{ {}_{p_1^{E,n}} }
    \\
    S^{4n+3}
    &\underset{h^{4n+3}_{\mathbb{H}}}{\longrightarrow}&
    \mathbb{H}P^n
    \,.
  }
$$

Moreover, by the defining [[universal property]] of the [[homotopy pushout]], the extension $p_1^{E,n+1}$ of $p_1^{E,n}$ is equivalently a choice of [[homotopy]] which trivializes the pullback of $p_1^{E,n}$ to the [[n-sphere|4n+3-sphere]]:

$$
  \array{
    \ast
    &
    \overset{}{\longrightarrow}
    &
    \Omega^{\infty - 4} E
    \\
    \big\uparrow
    &
    {}_{  p_1^{E,n+1}  }
    \seArrow
    &
    \big\uparrow \mathrlap{ ^{_{ p_1^{E,n} }} }
    \\
    S^{4n+3}
    &\underset{ h^{4n+3}_{\mathbb{H}} }{\longrightarrow}&
    \mathbb{H}P^n
    \,.
  }
$$

This means, first of all, that the non-triviality of the pullback class

$$
  \big(
    h^{4n+3}_{\mathbb{C}}
  \big)^\ast
  ( p_1^{E,n} )
  \;\in\;
  \widetilde E^4
  \big(
    S^{4n+3}
  \big)
  \;\simeq\;
  E_{4n - 1}
$$

is the [[obstruction]] to the existence of the extension/orientation at this stage.

It follows that if these obstructions all vanish, then a quaternionic $E$-orientation does exist.  A sufficient condition for this is, evidently, that the [[reduced cohomology|reduced]] $E$-cohomology of all $(4n-1)$-dimensional spheres vanishes.

=--

Hence:

+-- {: .num_prop #QuaternionicOrientationExistsIf4nMinus1DegreesAreTrivial}
###### Proposition

If $E$ is a [[multiplicative cohomology theory|multiplicative]]
[[Whitehead generalized cohomology theory]] whose [[graded-commutative ring|graded]] [[cohomology ring]] $E_\bullet$ is trivial in degrees $4 n - 1 $, then $E$ admits a quaternionic orientation (Def. \ref{QuaternionicOrientedCohomologyTheory}).

=--

## Examples
 {#Examples}

* [[ordinary cohomology]] has coefficient ring concentrated in degree 0, and hence satisfies the sufficient condition of Prop. \ref{QuaternionicOrientationExistsIf4nMinus1DegreesAreTrivial} to admit quaternionic orientation;

* [[KO-theory]] (orthogonal [[topological K-theory]]) has coefficient ring

  $$
    KO_n
    \,\simeq\,
    \left\{
    \array{
      \mathbb{Z} &\vert& n = 0 \,mod\, 8
      \\
      \mathbb{Z}_2 &\vert& n = 1 \,mod\, 8
      \\
      \mathbb{Z}_2 &\vert& n = 2 \,mod\, 8
      \\
      0 &\vert& n = 3 \,mod\, 8
      \\
      \mathbb{Z} &\vert& n = 4 \,mod\, 8
      \\
      0 &\vert& n = 5 \,mod\, 8
      \\
      0 &\vert& n = 6 \,mod\, 8
      \\
      0 &\vert& n = 7 \,mod\, 8
    }
    \right.
    \,,
  $$

  and hence satisfies the sufficient condition of Prop. \ref{QuaternionicOrientationExistsIf4nMinus1DegreesAreTrivial} to admit quaternionic orientation;

  (see also [Laughton 08, Example 2.2.7](#Laughton08))

* Among quaternionic oriented cohomology theories, [[quaternionic cobordism]] is universal in the fashion analogous to the [[universal complex orientation on MU]] ([Laughton 08, Example 2.2.9](#Laughton08))

## Related concepts

* [[MSp]]

## References

* {#ConnerFloyd66} [[Pierre Conner]], [[Edwin Floyd]], Chapter II, within Theorem 7.5 of: _[[The Relation of Cobordism to K-Theories]]_, Lecture Notes in Mathematics __28__ Springer 1966 ([doi:10.1007/BFb0071091](https://link.springer.com/book/10.1007/BFb0071091), [MR216511](http://www.ams.org/mathscinet-getitem?mr=216511))

* {#Baker92} [[Andrew Baker]], _Some chromatic phenomena in the homotopy of $MSp$_, in: N. Ray, G. Walker (eds.), _Adams Memorial Symposium on Algebraic Topology, Vol. 2_ editors, Cambridge University Press (1992), 263–80 ([pdf](http://www.maths.gla.ac.uk/~ajb/dvi-ps/bpmsp.pdf), [[BakerMSp.pdf:file]])

* [[Dai Tamaki]], [[Akira Kono]], Section 3.9 in: _Generalized Cohomology_, Translations of Mathematical Monographs, American Mathematical Society, 2006 ([ISBN: 978-0-8218-3514-2](https://bookstore.ams.org/mmono-230))

* {#Laughton08} [[Craig Laughton]], Example 2.2.5 in: _Quasitoric manifolds and cobordism theory_, Manchester 2008 ([pdf](http://www.ma.man.ac.uk/~nige/craiglphd.pdf), [[LaughtonQuasitoric.pdf:file]])

* [[Ivan Panin]], [[Charles Walter]], _Quaternionic Grassmannians and Pontryagin classes in algebraic geometry_, 2010 ([hal:00531725](https://hal.archives-ouvertes.fr/hal-00531725), [pdf](https://www.math.uni-bielefeld.de/LAG/man/405.pdf))

* [[Ivan Panin]], [[Charles Walter]], _Quaternionic Grassmannians and Borel classes in algebraic geometry_ ([arXiv:1011.0649](https://arxiv.org/abs/1011.0649))

* [[Ivan Panin]], [[Charles Walter]], _On the algebraic cobordism spectra $MSL$ and $MSp$_ ([arXiv:1011.0651](https://arxiv.org/abs/1011.0651))

* [[Ivan Panin]], _[Oriented theories and symplectic cobordism](https://www.esaga.uni-due.de/marc.levine/SpecialSemester/Panin/)_, Seminar

[[!redirects quaternionic oriented cohomology theories]]

[[!redirects quaternionic oriented cohomology]]


quaternionic projective plane

#Contents#
* table of contents
{:toc}

## Idea

The _quaternionic projective plane_ $\mathbb{H}P^2$ is the [[projective plane]] over the [[skew-field]] of [[quaternions]].

## Properties

### As a quaternion-Kähler manifold / Wolf space

The quaternionic projective plane $\mathbb{H}P^2$ the first of the list of examples  of spaces that are both [[quaternion-Kähler manifolds]] as well as [[symmetric spaces]], called _[[Wolf spaces]]_.

### AKM-Theorem

In higher dimensional analogy to the [[Arnold-Kuiper-Massey theorem]] identifying the [[quotient space|quotient]] of the [[complex projective plane]] by its [[finite group of order 2|O(1)]]-action as the [[4-sphere]], we have:

+-- {: .num_prop}
###### Proposition
**(quaternionic [[AKM-theorem]])**

The [[7-sphere]] is the [[quotient space]] of the (right-)[[quaternionic projective plane]] by the left multiplication [[action]] by [[U(1)]] $\subset$ [[Sp(1)]]:

$$
  \mathbb{H}P^2 / \mathrm{U}(1)
  \simeq
  S^7
$$

=--

([Arnold 99](#Arnold99), [Atiyah-Witten 01, Sec. 5.5](#AtiyahWitten01))

## Related concepts

* [[quaternionic projective space]]

  * [[quaternionic projective line]]

* [[real projective plane]]

* [[complex projective plane]]

* [[octonionic projective plane]]

## References

* {#Arnold99} [[Vladimir Arnold]], _Relatives of the Quotient of the Complex Projective Plane by the Complex Conjugation_, Tr. Mat. Inst. Steklova, 1999, Volume 224, Pages 56–67; English translation: Proceedings of the Steklov Institute of Mathematics, 1999, 224, 46–56 ([mathnet:tm691](http://mi.mathnet.ru/eng/tm691))

{#AtiyahWitten01} [[M-theory on 8-manifolds|M-theory on the 8-manifold]] $\mathbb{H}P2$, hence on a [[quaternion-Kähler manifold]] of dimension 8 with [[holonomy]] [[Sp(2).Sp(1)]], is considered in

* [[Michael Atiyah]], [[Edward Witten]], p. 75 onwards in  _$M$-Theory dynamics on a manifold of $G_2$-holonomy_, Adv. Theor. Math. Phys. 6 (2001) ([arXiv:hep-th/0107177](http://arxiv.org/abs/hep-th/0107177))

and argued to be [[duality in string theory|dual]] to [[M-theory on G2-manifolds]] in three different ways, which in turn is argued to lead to a a possible [[proof]] of [[confinement]] in the resulting 4d [[effective field theory]] (see [there](M-theory+on+G2-manifolds#Confinement) for more).

[[!redirects HP^2]]
[[!redirects HP2]]


quaternionic projective space

#Contents#
* table of contents
{:toc}

## Idea

The [[quaternion|quaternionic]] [[projective space]] $\mathbb{H}P^n$ is the [[space]] of right (or left) [[quaternion]] [[lines]] through the origin in $\mathbb{H}^{n+1}$, hence the space of [[equivalence classes]] $[q_1, \cdots, q_{n+1}]$ of [[n-tuple|(n+1)-tuples]]  of [[quaternions]] excluding zero, under the [[equivalence relation]] given by right (or left) [[multiplication]] with non-zero quaternions

$$
  \mathbb{H}P^n
  \;\coloneqq\;
  \big\{
    [q_1, \cdots, q_{n+1}]
  \big\}
  \;\coloneqq\;
  \Big(
    \big\{
      (q_1, \cdots, q_{n+1})
    \big\}
    \setminus
    \{(0, \cdots, 0)\}
  \Big)
  /_{
    (q_1, \cdots, q_{n+1})
      \sim
    (q_1 q, \cdots, q_{n+1} q)
    \vert
    q \neq 0
  }
$$

## Properties
 {#Properties}

### General

\begin{proposition}
Every continuous map $\mathbb{H}P^n\rightarrow\mathbb{H}P^n$ for $n \geq 2$ has a [[fixed point]]. This does not hold for $n = 1$ as in this case $\mathbb{H}P^1\cong S^4$ and the antipodal map $S^4\rightarrow S^4, x\mapsto -x$ does not have a fixed point.
\end{proposition}

([Hatcher 02, page 180](#Hatcher02))

### Homology

+-- {: .num_prop #HomologyAndCohomologyOfQuaternionicProjectiveSpace}
###### Proposition
**([[homology]] of quaternionic projective space)**

The [[ordinary homology]] [[homology groups|groups]] of quaternionic projective space $\mathbb{R}P^n$ can be calculated using its CW structure and are given by

\[
  \label{ListOfHomotopyGroupsOfQuaternionicProjectiveSpace}
    H^k
    \big(
      \mathbb{H}P^n
    \big)
    \;=\;
    \left\{
    \array{
       \mathbb{Z} &\vert& \; k = 0, 4,\ldots, 4(n-1),4n
       \\
       1 &\vert& otherwise
    }
    \right.
\]

=--

### As a coset space

As any [[Grassmannian]], quaternion projective space is canonically a [[coset space]], in this case of the [[quaternion unitary group]] $Sp(n+1)$ by the [[central product group]] [[Sp(n).Sp(1)]]:

\[
  \label{SpnCosetSpaceRealization}
  \mathbb{H}P^n
  \;\simeq\;
  \frac{
    Sp(n+1)
  }{
    Sp(n)\cdot Sp(1)
  }
\]

### As a quaternion-Kähler symmetric space (Wolf space)

By the [[coset space]]-realization (eq:SpnCosetSpaceRealization), quaternion projective space is naturally a [[quaternion-Kähler manifold]] which is also a [[symmetric space]]. As such it is an example of a [[Wolf space]].

## Related concepts

* [[real projective space]]

* [[complex projective space]]

* [[octonionic projective space]]

## References

### General

* {#Hatcher02} [[Allen Hatcher]], *Algebraic Topology*, Cambridge University Press (2002) [[ISBN:9780521795401](https://www.cambridge.org/gb/academic/subjects/mathematics/geometry-and-topology/algebraic-topology-1?format=PB&isbn=9780521795401), [webpage](https://pi.math.cornell.edu/~hatcher/AT/ATpage.html)]

* {#ConnerFloyd66} [[Pierre Conner]], [[Edwin Floyd]], Section I.4 of: _[[The Relation of Cobordism to K-Theories]]_, Lecture Notes in Mathematics __28__ Springer 1966 ([doi:10.1007/BFb0071091](https://link.springer.com/book/10.1007/BFb0071091), [MR216511](http://www.ams.org/mathscinet-getitem?mr=216511))

See also

* [[Topospaces]], _[Homology of quaternionic projective space](https://topospaces.subwiki.org/wiki/Homology_of_quaternionic_projective_space)_

* Wikipedia, _[Quaternionic projective space](https://en.wikipedia.org/wiki/Quaternionic_projective_space)_

### In string theory
  {#ReferencesInStringTheory}

[[M-theory on 8-manifolds|M-theory on the 8-manifold]] $\;$ [[HP2]], hence on a [[quaternion-Kähler manifold]] of dimension 8 with [[holonomy]] [[Sp(2).Sp(1)]], is considered in

* [[Michael Atiyah]], [[Edward Witten]], p. 75 onwards in  _$M$-Theory dynamics on a manifold of $G_2$-holonomy_, Adv. Theor. Math. Phys. 6 (2001) ([arXiv:hep-th/0107177](http://arxiv.org/abs/hep-th/0107177))

and argued to be [[duality in string theory|dual]] to [[M-theory on G2-manifolds]] in three different ways, which in turn is argued to lead to a a possible [[proof]] of [[confinement]] in the resulting 4d [[effective field theory]] (see [there](M-theory+on+G2-manifolds#Confinement) for more).

[[!redirects quaternionic projective spaces]]

[[!redirects HP^n]]
[[!redirects HPn]]

[[!redirects HP^1]]
[[!redirects HP1]]
[[!redirects quaternionic projective line]]
[[!redirects quaternionic projective lines]]


quaternionic quantum mechanics

#Contents#
* table of contents
{:toc}

## Idea

_Quaternionic quantum mechanics_ is the proposal to consider the principles of [[quantum mechanics]] but with the [[ground field]] of [[complex numbers]] systematically replaced by the ring of [[quaternions]] instead.

## References

* [[Stephen Adler]], _Quaternionic Quantum Mechanics and Quantum Fields_, International series of monographs on physics 88 Oxford University Press.
* [[Valter Moretti]], Marco Oppio  Quantum theory in quaternionic Hilbert space: How Poincaré symmetry reduces the theory to the standard complex one
Rev. Math. Phys. 31, (2019) 1950013


quaternionic set

+-- {: .rightHandSide}
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###Context###
#### Topos theory
+--{: .hide}
[[!include topos theory - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A **quaternionic set** is a [[simplicial set]] with a [[quaternion group|quarternionic]] symmetry, in other words it is a [[presheaf]] on the quaternionic simplex category $\Delta Q$ which contains the ordinary [[simplex category]] $\Delta$ and an opposite [[quaternion group]] $Q_{n+1}^{op}$ as the automorphism group at the objects $[n]$, whence quaternionic sets are an example of an [[skew-simplicial set|crossed simplicial object]].

## Definition

Let $Q_n$ be the [[group]] given by the [[group presentation|presentation]] $\{x,y| x^n  = y^2\;,y x y^{-1}=x^{-1}\}$, hence the $n$th [[quaternion group]], also called a *[[binary dihedral group]]*.

\begin{definition}\label{Definition}
The **quaternionic simplex category**[^term] $\Delta Q$ has objects $[n],\; n\geq 0\;$, contains the (topologists’) [[simplex category]] $\Delta$ as a (non-[[full subcategory|full]]) [[wide subcategory]] and each [[object]] $[n]$ has as [[automorphism group]] the ([[opposite group|opposite]] of the) $n+1$st [[quaternion group]]: $Aut_{\Delta Q}([n]) = Q_{n+1}^{op}$.

Furthermore, any [[morphism]] $\varphi:[m]\to [n]$ in $\Delta Q$ can be uniquely factored as $\varphi = s\circ q$ with $s:[m]\to [n]\in \Delta$ and $q\in Q_{m+1}^{op}$.

A **quaternionic set** is a [[presheaf]] on $\Delta Q$.
\end{definition}

[^term]: Since $\Delta Q$ encapsulates exactly the structure of $Q_*$ (i.e. the simplicial set that has fibers $Q_{n+1}$ at $[n]$ together with simplicially compatible actions) it is also referred to as the _quaternionic crossed simplicial group_. The terminology used at [[crossed group]] would refer to it as the _total category_ of the [[skew-simplicial set|crossed simplicial group]] $Q_*$. Note also that we follow [Loday](#Loday87) by taking $Q_n^{op}$ as automorphism group at $[n-1]$ whereas other authors like Krasauskas or Dyckerhoff-Kapranov use $Q_n$.

## Properties

* According to the classification of [[crossed group|crossed simplicial groups]], $\Delta Q$ or, more precisely, the equivalent crossed group $Q_*$ corresponds to the [[exact sequence]] of crossed groups $1\to \mathbb{Z}/2\to Q_*\to D_*\to 1\; .$ Whence it is of _dihedral type_ and, accordingly, the [[model category]] techniques for planar crossed simplicial groups apply (cf. [Spaliński](#Spalinski95), [Balchin](#Balchin18)).

* The [[geometric realization]] of $Q_*$ is the normalizer of $S^1$ in $S^3$ (cf. [Loday](#Loday87)).

* Just like in the [[cyclic set|cyclic]] and [[dihedral set|dihedral]] cases, $\Delta Q$ is _self-dual_: $\Delta Q\cong \Delta Q^{op}$ (cf. [Dunn](#Dunn89), prop.1.4).

* As usual for [[crossed group|crossed groups]] (or, more generally, for functors between small categories that are surjective on objects; cf. [[Sheaves in Geometry and Logic|Moerdijk-Maclane]], pp.359, 377f), the inclusion $i:\Delta\to \Delta Q$ induces by [[Kan extension]] an [[essential geometric morphism|essential]] [[surjective geometric morphism]] $i_!\vdash i^*\vdash i_*: Set^{\Delta^{op}}\to Set^{\Delta Q^{op}}\;$. Here the [[inverse image]] $i^*:Set^{\Delta Q^{op}}\to Set^{\Delta^{op}}$ restricts a presheaf $P$ on $\Delta Q$ to one on $\Delta$ by pre-composition: $i^*(P)=P\circ i^{op}\,$.

* Since $i:Set^{\Delta^{op}}\to Set^{\Delta Q^{op}}$ is surjective, it follows again by generalities that $Set^{\Delta Q^{op}}$ is (equivalent to) the category of algebras for the [[monad]] induced by $i^*i_!$ on $Set^{\Delta^{op}}$ whence quaternionic sets are indeed [[simplicial set|simplicial sets]] enhanced with a quaternionic algebra structure. From this perspective, $i_!\vdash i^*$ is an [[adjunction]] between a [[free functor|free algebra functor]] and an underlying (simplicial) set functor. (For an explicit description of the free [[skew-simplicial set]] on a [[simplicial set]] and the corresponding monad for general crossed groups $G_*$ see [Krasauskas](#Krasauskas87).)

* $\Delta Q$ is an [[Eilenberg-Zilber category]] with degree function $deg:\Delta Q\to \mathbb{N}$ inherited from $\Delta$. The same holds for other types of [[skew-simplicial set|skew simplicial sets]], or more generally, for [[crossed group|crossed groups]] over a strict [[Reedy category]] (cf. [Berger-Moerdijk](#BM11) ex. 6.8).

* Truncating $\Delta Q$ at $[1]$ and taking presheaves, one obtains a _category of quaternionic graphs_ $Set^{\Delta Q_{\leq 1}^{op}}$ with a $Q_2$ action, just like [[graphic category|reflexive graphs]] result from $\Delta_1$, resp. reversible graphs from truncating $\Delta S$ in the [[symmetric set|symmetric simplex category]].

## Related entries

* [[skew-simplicial set]]

* [[crossed group]]

* [[cyclic set]]

* [[dihedral set]]

* [[quaternion]]

## References

* {#Loday87}[[Jean-Louis Loday]], _Homologies diédrale et quaternionique_, Adv. in Math. **66** (1987) pp.119-148.

* {#Dunn89}Gerald Dunn, _Dihedral and Quaternionic Homology and Mapping Spaces_, K-theory **3** (1989) pp.141-161.

* {#Spalinski95}[[Jan Spalinski|Jan Spaliński]], _Homotopy theory of dihedral and quaternionic sets_, Topology **39** (1995) pp.35-52.

* {#Krasauskas87}R. Krasauskas, _Skew-simplicial groups_, Lithuanian Math. J. **27** pp.47-54 (1987).

* {#FL91}Zbigniew Fiedorowicz, [[Jean-Louis Loday]], _Crossed simplicial groups and their associated homology_, Trans. AMS **326** pp.57-87 (1991).

*  {#BM11}[[Clemens Berger]], [[Ieke Moerdijk]], _On an extension of the notion of Reedy category_, Math. Z. **269** (2011) pp.977-1004. ([arXiv:0809.3341](http://arxiv.org/abs/0809.3341))

* [[Tobias Dyckerhoff]], [[Mikhail Kapranov]], _Crossed simplicial groups and structured surfaces_, arXiv:1403.5799 (2014). ([abstract](https://arxiv.org/abs/1403.5799))

* {#Balchin18}[[Scott Balchin]], _Homotopy of planar Lie group equivariant presheaves_, J. Homotopy Relat. Struct. **13** (2018) pp.555-580. ([doi](https://doi.org/10.1007/s40062-017-0193-z))


quaternionic structure

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Almost quaternionic structure: [[reduction of structure group|reduction of the stucture group]] along the inclusion $GL_n(\mathbb{H}) \hookrightarrow Gl_{4 n}(\mathbb{R})$, for $\mathbb{H}$ the [[quaternions]].

(...)

## Related concepts

* [[quaternionic unitary group]]

* [[real structure]], [[complex structure]]

* [[real vector space]], [[complex vector space]], [[quaternionic vector space]]

* [[para-quaternionic structure]]

## References

* Wikipedia, _[Quaternionic structure](http://en.wikipedia.org/wiki/Quaternionic_structure)_

* Wikipedia, _[Reduction of the structure group -- Examples](https://en.wikipedia.org/wiki/Reduction_of_the_structure_group#Examples)_

[[!redirects quaternionic structures]]

[[!redirects almost quaternionic structure]]


quaternionic unitary group

> See also _[[compact symplectic group]]_.

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

This Lie group is the analog of the [[unitary group]] as one passes from the [[complex numbers]] to the [[quaternions]].

The quaternionic unitary group $Sp(n)$ is the group of quaternion-unitary transformations of $\mathbb{H}^n$.  It is also called the [[compact symplectic group]], since both it and the [[symplectic group]] $Sp(2n, \mathbb{R})$ are real forms of the complex Lie group $Sp(2n,\mathbb{C})$, and it is the compact form.

## Properties

### Exceptional isomorphisms

* $Sp(1) \simeq$ [[Spin(3)]] ([this Prop.](Spin3#ExceptionalIsoToSp2))

* $Sp(2) \simeq$ [[Spin(5)]] ([this Prop.](Spin5#ExceptionalIsomorphismOfSU2ToSpin3AndSp1))

## Related concepts

* A [[Riemannian manifold]] of [[dimension]] $4n$ is called a _[[quaternion-Kähler manifold]]_ if its [[holonomy group]] is a [[subgroup]] of the [[quotient group]] [[Sp(n).Sp(1)]] of the [[direct product group]] $Sp(n) \times Sp(1)$. If it is even a subgroup  of just the $Sp(n)$ factor, then it is called a _[[hyperkähler manifold]]_.

* [[Gromoll-Meyer sphere]]

* [[quaternionic projective space]]

* [[symplectic group]]

* [[SL(2,H)]]

## References

* [[Howard Georgi]], §26 in: *Lie Algebras In Particle Physics*, Westview Press (1999), CRC Press (2019) [[doi:10.1201/9780429499210](https://doi.org/10.1201/9780429499210)]

* _Quaternionic groups_ ([pdf](http://www-math.mit.edu/~dav/quatcoordfree.pdf))

[[!redirects quaternionic unitary groups]]

[[!redirects Sp(n)]]

[[!redirects quaternion unitary group]]
[[!redirects quaternion unitary groups]]


quaternionic vector bundle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[vector bundle]] with [[fibers]] [[quaternionic vector spaces]].

## Related concepts

* [[real vector bundle]], [[complex vector bundle]]

* [[quaternionic oriented cohomology theory]]

* [[quaternionic K-theory]]

## References

### General

* Semyon Alesker, _A topological obstruction to existence of quaternionic Plucker map_, in: _Geometric Aspects of Functional Analysis_, Lecture Notes in Mathematics, vol 1850. Springer, ([arXiv:math/0111028](https://arxiv.org/abs/math/0111028), [doi:10.1007/978-3-540-44489-3_1](https://doi.org/10.1007/978-3-540-44489-3_1))

### Quaternionic line bundles

* [[Gustavo Granja]], _On quaternionic line bundles_, MIT 1999 ([dspace:1721.1/85302](https://dspace.mit.edu/handle/1721.1/85302), [pdf](https://dspace.mit.edu/bitstream/handle/1721.1/85302/43854510-MIT.pdf?sequence=2&isAllowed=y))

* Diarmuid Crowley, [[Sebastian Goette]], _Kreck-Stolz invariants for quaternionic line bundles_, Transactions of the American Mathematical Society
Vol. 365, No. 6 (2013), pp. 3193-3225 ([arXiv:1012.5237](https://arxiv.org/abs/1012.5237), [jstor:23513538](https://www.jstor.org/stable/23513538))

Relation to 4-[[cohomotopy]] over [[5-manifolds]]:

* Panagiotis Konstantis, _Vector bundles and cohomotopies of spin 5-manifolds_, Homology, Homotopy and Applications Volume 23 (2021) Number 1 ([arXiv:1812.06547](https://arxiv.org/abs/1812.06547), [doi:10.4310/HHA.2021.v23.n1.a9](https://dx.doi.org/10.4310/HHA.2021.v23.n1.a9))

[[!redirects quaternionic vector bundles]]

[[!redirects quaternionic line bundle]]
[[!redirects quaternionic line bundles]]


quaternionic vector space

#Contents#
* table of contents
{:toc}

## Idea

A _quaternionic vector space_ is a [[module]] over the [[quaternions]].

## Related concepts

* [[real vector space]], [[complex vector space]]

* [[quaternionic vector bundle]]

## References

See also

* Wikipedia, _[Quaternionic vector space](https://en.m.wikipedia.org/wiki/Quaternionic_vector_space)_

[[!redirects quaternionic vector spaces]]


qudit

+-- {: .rightHandSide}
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### Context
#### Computation
+-- {: .hide}
[[!include constructivism - contents]]
=--
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum information theory]] and [[quantum computing]], by a *qdit* (or *qudit*) one means a [[quantum state]] in a $d$-[[dimension of a vector space|dimensional]] [[Hilbert space]], for any [[natural number]] $d$.

Hence for fixed $d\in \mathbb{N}$, the quantum [[data type]] of qdits is the $d$-[[dimension of a vector space|dimensional]] [[complex vector space]] equipped with a [[quantum measurement]]-[[linear basis|basis]]

$$
  \mathbb{C}^d
  \;\simeq\;
  \underset{n \in \{1, \cdots, d\}}{\bigoplus} \mathbb{C} \cdot \vert n \rangle
  \,.
$$

* For $d = 2$ one speaks of *[[qbits]]*, this is the original terminology;

* For $d = 3$ one also speaks of *qtrits*.

## Examples

### Anyon states
 {#AnyonStates}

A key example of [[quantum gates]] which naturally act on qdits for $d \gt 2$ are [[anyon]] [[braid representation|braid]]-gates in [[topological quantum computation]]:

For the potentially realistic case of [[Chern-Simons theory]]/[[WZW-model]]-controled [[anyons]] (such as [[su(2)-anyons]]), the elementary [[quantum logic gates]] act by the [[monodromy]] of the [[Knizhnik-Zamolodchikov connection]] on [[Hilbert spaces]] of [[conformal blocks]], whose [[dimension of a vector space|dimension]] $d$ is given by a [[Verlinde formula]].

References which make this point explicit include [Kolganov, Mironov & Morozov (2023)](#KolganovMironovMorozov23).

## Related concepts

* [[data type]]

* [[qbit]]

* [[quantum gate]], [[quantum circuit]]

## References
 {#References}

General:

* IEEE Spectrum *[Qudits: The Real Future of Quantum Computing?](https://spectrum.ieee.org/qudits-the-real-future-of-quantum-computing)* (June 2017)

* Yuchen Wang, Zixuan Hu, Barry C. Sanders, Sabre Kais, *Qudits and High-Dimensional Quantum Computing*, Front. Phys. (2020) [[doi:10.3389/fphy.2020.589504](https://doi.org/10.3389/fphy.2020.589504)]

See also:

* Wikipedia, *[Qutrit](https://en.wikipedia.org/wiki/Qutrit)*

Explicit mentioning of the qdit-nature of the elementary gates in [[topological quantum computation]]:

* {#KolganovMironovMorozov23}  [[Nikita Kolganov]], [[Sergey Mironov]], [[Andrey Morozov]], *Large $k$ topological quantum computer*, Nuclear Physics B
**987** (2023) 116072 [[arXiv:2105.03980](https://arxiv.org/abs/2105.03980), [doi:10.1016/j.nuclphysb.2023.116072](https://doi.org/10.1016/j.nuclphysb.2023.116072)]

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###Context###
#### Cobordism theory
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[[!include cobordism theory -- contents]]
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This entry is about the article

* [[René Thom]]:

  \linebreak

  **Quelques propriétés globales des variétés différentiables**

  \linebreak

  Comment. Math. Helv. 28, (1954). 17-86

  [doi:10.1007/BF02566923](https://doi.org/10.1007/BF02566923)

  [dml:139072](https://eudml.org/doc/139072)

  [digiz:GDZPPN002056259](http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002056259)

  [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/thomcob.pdf)

on [[differential topology]], proving [[Thom's theorem]] which identifies [[cobordism classes]] of [[normally oriented submanifolds]] with [[homotopy classes]] of maps into a universal [[Thom space]] $M SO(n)$.

Together with

* {#Pontrjagin38} [[Lev Pontrjagin]], _[[Classification of continuous maps of a complex into a sphere]]_, Dokl. Akad. Nauk SSSR 19 (1938), 361-363

or rather its belated exposition in

* {#Pontrjagin55} [[Lev Pontrjagin]], _[[Smooth manifolds and their applications in homotopy theory]]_, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955, AMS Translation Series 2, Vol. 11, 1959 ([doi:10.1142/9789812772107_0001](https://www.worldscientific.com/doi/abs/10.1142/9789812772107_0001), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/pont001.pdf))

due to which Thom's construction came to be mainly known as the _[[Pontryagin-Thom construction]]_, this lays the foundations of [[cobordism theory]] as such and as a tool in [[stable homotopy theory]].

#Contents#
* table of contents
{:toc}

## Chapter I -- Properties of differentiable maps

### 1. Definitions

* [[critical point]], [[regular point]] (p. 18)

### 3. Pre-image of a sub-manifold

* [[tubular neighbourhood]] (p. 21)

### 4. Pre-image of a sub-manifold under a t-regular map

* [[Thom's transversality theorem]] (p. 26)

## Chapter II -- Sub-manifolds and homology classes of a manifold

### 2. Complex associated to a closed subgroup of the orthogonal group

* [[Thom space]] (p. 29)

## Chapter III -- On a problem of Steenrod

## Chapter IV -- Cobordant differentiable manifolds

* [[bordism]], [[cobordism]] (p. 65)

### 3.

### 4. Cobordant sub-manifolds

* [[normally oriented submanifold]] (p. 67)

* [[Pontryagin-Thom collapse]] (p. 69)

* [[cobordism classes]] of [[submanifolds]] (p. 71)

### 5. A fundamental theorem

* [[Thom's theorem]] (p. 74, 75)

category: reference