higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
function algebras on ∞-stacks?
derived smooth geometry
The coset space of Spin(9) by Spin(7) is homeomorphic to the 15-sphere:
This coset realization features in the octonionic Hopf fibration, making manifest its $Spin(9)$-equivariance (Ornea-Parton-Piccinni-Vuletescu 12, p. 7).
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)
Alfred Gray, Paul S. Green, p. 2 of Sphere transitive structures and the triality automorphism, Pacific J. Math. Volume 34, Number 1 (1970), 83-96 (euclid:1102976640)
Liviu Ornea, Maurizio Parton, Paolo Piccinni, Victor Vuletescu, Spin(9) geometry of the octonionic Hopf fibration, Transformation Groups (2013) 18: 845 (arXiv:1208.0899, doi:10.1007/s00031-013-9233-x)
Last revised on April 29, 2019 at 04:41:03. See the history of this page for a list of all contributions to it.