higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
function algebras on ∞-stacks?
derived smooth geometry
The coset space of the exceptional Lie group G2 by its special unitary subgroup SU(3) is diffeomorphic to the 6-sphere:
coset space-structures on n-spheres:
standard: | |
---|---|
$S^n \simeq_{diff} SO(n+1)/SO(n)$ | this Prop. |
$S^{2k+1} \simeq_{diff} SU(k+1)/SU(k)$ | this Prop. |
exceptional: | |
$S^7 \simeq_{top} Spin(7)/G_2$ | Spin(7)/G2 is the 7-sphere |
$S^7 \simeq_{top} Spin(6)/SU(3)$ | since Spin(6) $\simeq$ SU(4) |
$S^7 \simeq_{top} Spin(5)/SU(2)$ | Spin(5)/SU(2) is the 7-sphere |
$S^6 \simeq_{top} G_2/SU(3)$ | G2/SU(3) is the 6-sphere |
see also Spin(8)-subgroups and reductions
homotopy fibers of homotopy pullbacks of classifying spaces:
(from FSS 19, 3.4)
According to ABF 17 the statement is originally due to to
Review:
Robert Bryant, Section 1.3 of Manifolds with $G_2$-Holonomy, lecture notes, 1998 (web)
Robert Bryant, Section 2.3 of Some remarks on $G_2$-structures, Proceedings of 12th Gokova Geometry-Topology Conference, 2005 (pdf)
Simon Salamon, p. 6 of A tour of exceptional geometry, Milanj. math.72 (2002)1-0 (web)
A. J. MacFarlane, The sphere $S^6$ viewed as a $G_2/SU(3)$-coset space, International Journal of Modern Physics A Vol. 17, No. 19, pp. 2595-2613 (2002) (doi:10.1142/S0217751X02010650)
A. Gyenge, Section 4.2 of On the topology of the exceptional Lie group $G_2$, 2011 (pdf)
Ilka Agricola, Aleksandra Borówka, Thomas Friedrich, $S^6$ and the geometry of nearly Kähler 6-manifolds (arXiv:1707.08591)
Last revised on April 19, 2019 at 16:19:46. See the history of this page for a list of all contributions to it.