quaternionic projective line$\,\mathbb{H}P^1$
This is one of the parallelizable spheres, as such corresponds to the octonions among the division algebras, being the manifold of unit octonions, and is the only one of these which does not carry (Lie) group structure but just Moufang loop structure.
The 7-sphere participates in the quaternionic Hopf fibration, the analog of the complex Hopf fibration with the field of complex numbers replaced by the division ring of quaternions or Hamiltonian numbers $\mathbb{H}$.
Here the idea is that $S^7$ can be construed as $\{(x, y) \in \mathbb{H}^2: {|x|}^2 + {|y|}^2 = 1\}$, with $p$ mapping $(x, y)$ to $x/y$ as an element in the projective line $\mathbb{P}^1(\mathbb{H}) \cong S^4$, with each fiber a torsor parametrized by quaternionic scalars $\lambda$ of unit norm (so $\lambda \in S^3$). This canonical $S^3$-bundle (or $SU(2)$-bundle) is classified by a map $S^4 \to \mathbf{B} SU(2)$.
(coset space of Spin(7) by G2 is 7-sphere)
Consider the canonical action of Spin(7) on the unit sphere in $\mathbb{R}^8$ (the 7-sphere),
This action is is transitive;
the stabilizer group of any point on $S^7$ is G2;
all G2-subgroups of Spin(7) arise this way, and are all conjugate to each other.
Hence the coset space of Spin(7) by G2 is the 7-sphere
(e.g Varadarajan 01, Theorem 3)
Other coset realizations of the usual differentiable 7-sphere (Choquet-Bruhat, DeWitt-Morette 00, p. 288):
$S^7 \simeq_{diff}$ Spin(6)$/SU(3) \simeq_{iso} SU(4)/SU(3)$ (by this Prop.);
$S^7 \simeq_{diff} Spin(5)/SU(2)$ (Awada-Duff-Pope 83, Duff-Nilsson-Pope 83)
These three coset realizations of βsquashedβ 7-spheres together with a fourth
the realization of the βroundβ 7-sphere, may be seen jointly as resulting from the 8-dimensional representations of even Clifford algebras in 5, 6, 7, and 8 dimensions (see Baez) and as such related to the four normed division algebras. See also Choquet-Bruhat+DeWitt-Morette00, pp. 263-274.
coset space-structures on n-spheres:
standard: | |
---|---|
$S^{n-1} \simeq_{diff} SO(n)/SO(n-1)$ | this Prop. |
$S^{2n-1} \simeq_{diff} SU(n)/SU(n-1)$ | this Prop. |
$S^{4n-1} \simeq_{diff} Sp(n)/Sp(n-1)$ | this Prop. |
exceptional: | |
$S^7 \simeq_{diff} Spin(7)/G_2$ | Spin(7)/G2 is the 7-sphere |
$S^7 \simeq_{diff} Spin(6)/SU(3)$ | since Spin(6) $\simeq$ SU(4) |
$S^7 \simeq_{diff} Spin(5)/SU(2)$ | since Sp(2) is Spin(5) and Sp(1) is SU(2), see Spin(5)/SU(2) is the 7-sphere |
$S^6 \simeq_{diff} G_2/SU(3)$ | G2/SU(3) is the 6-sphere |
$S^15 \simeq_{diff} Spin(9)/Spin(7)$ | Spin(9)/Spin(7) is the 15-sphere |
see also Spin(8)-subgroups and reductions
homotopy fibers of homotopy pullbacks of classifying spaces:
(from FSS 19, 3.4)
The following gives an exotic 7-sphere:
A celebrated result of Milnor is that $S^7$ admits exotic smooth structures (see at exotic 7-sphere), i.e., there are smooth manifold structures on the topological manifold $S^7$ that are not diffeomorphic to the standard smooth structure on $S^7$. More structurally, considering smooth structures up to oriented diffeomorphism, the different smooth structures form a monoid under a (suitable) operation of connected sum, and this monoid is isomorphic to the cyclic group $\mathbb{Z}/(28)$. With the notable possible exception of $n = 4$ (where the question of existence of exotic 4-spheres is wide open), exotic spheres first occur in dimension $7$. This phenomenon is connected to the h-cobordism theorem (the monoid of smooth structures is identified with the monoid of h-cobordism classes of oriented homotopy spheres).
One explicit construction of the smooth structures is given as follows (see Milnor 1968). Let $W_k$ be the algebraic variety in $\mathbb{C}^5$ defined by the equation
and $S_\epsilon \subset \mathbb{C}^5$ a sphere of small radius $\epsilon$ centered at the origin. Then each of the $28$ smooth structures on $S^7$ is represented by an intersection $W_k \cap S_\epsilon$, as $k$ ranges from $1$ to $28$. These manifolds sometimes go by the name Brieskorn manifolds or Brieskorn spheres or Milnor spheres.
Let $\phi_0 \in \Omega^3(\mathbb{R}^7)$ be the associative 3-form and let
be given by
(where $x_0$ denotes the canonical coordinate on the first factor of $\mathbb{R}$ and $\phi_0$ is pulled back along the projection to $\mathbb{R}^7$) .
By construction this is its own Hodge dual
This implies that as we restrict $\Phi_0$ to
then there is a unique 3-form
on the 7-sphere such that
This 3-form $\phi$ defines a G2-structure on $S^7$. It is nearly parallel in that
(e.g. Lotay 12, def.2.4)
quaternionic projective line$\,\mathbb{H}P^1$
Martin Cederwall, Christian R. Preitschopf, The Seven-sphere and its Kac-Moody Algebra, Commun. Math. Phys. 167 (1995) 373-394 (arXiv:hep-th/9309030)
Takeshi Γno, On the Hopf fibration $S^7 \to S^4$ over $Z$, Nagoya Math. J. Volume 59 (1975), 59-64. (Euclid)
Relation to the Milnor fibration:
An ADE classification of finite subgroups of $SO(8)$ acting freely on $S^7$ (see at group action on an n-sphere) such that the quotient is spin and has at least four Killing spinors (see also at ABJM model) is in
Paul de Medeiros, JosΓ© Figueroa-O'Farrill, Sunil Gadhia, Elena MΓ©ndez-Escobar, Half-BPS quotients in M-theory: ADE with a twist, JHEP 0910:038,2009 (arXiv:0909.0163, pdf slides)
Paul de Medeiros, JosΓ© Figueroa-O'Farrill, Half-BPS M2-brane orbifolds (arXiv:1007.4761)
Discussion of subgroups:
Discussion of exotic smooth structures on 7-spheres includes
The explicit construction of exotic 7-spheres by intersecting algebraic varieties with spheres is described in
Discussion of (nearly) G2-structures on $S^7$ and calibrated submanifolds includes
On coset-realizations:
Linus Kramer, Octonion Hermitian quadrangles, Bull. Belg. Math. Soc. Simon Stevin Volume 5, Number 2/3 (1998), 353-362 (euclid:1103409015)
Y. Choquet-Bruhat, C. DeWitt-Morette, Analysis, Manifolds and Physics Part II, North Holland 2000
M. A. Awada, Mike Duff, Christopher Pope, $N=8$ Supergravity Breaks Down to $N=1$, Phys. Rev. Lett. 50, 294 β Published 31 January 1983 (doi:10.1103/PhysRevLett.50.294)
Mike Duff, Bengt Nilsson, Christopher Pope, Spontaneous Supersymmetry Breaking by the Squashed Seven-Sphere, Phys. Rev. Lett. 50, 2043 β Published 27 June 1983; Erratum Phys. Rev. Lett. 51, 846 (doi:10.1103/PhysRevLett.50.2043)
John Baez, Rotations in the 7th Dimension, (blog post), and TWF 195, (webpage)
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