synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
What is called the Gromoll-Meyer sphere (Gromoll-Meyer 74) is an example of an exotic 7-sphere. It arises as a double coset space/biquotient of the quaternionic unitary group Sp(2) $\simeq$ Spin(5) by two copies of Sp(1) $\simeq$ SU(2).
Consider the quaternion unitary group Sp(2) $\simeq$ Spin(5) and its two subgroup inclusion of Sp(1) $\simeq$ SU(2) as
Then the double coset space/biquotient $Sp(1)\backslash Sp(2) /Sp(1)$ with respect to the left and right multiplication action, respectively, of these two subgroups is, with respect to its canonically induced geometric structures:
homeomorphic to the topological 7-sphere;
diffeomorphic to an exotic 7-sphere.
As such, it is called the Gromoll-Meyer sphere, due to Gromoll-Meyer 74
The Gromoll-Meyer sphere is the only exotic 7-sphere that can be modeled by a biquotient of a compact Lie group (KZ02, Corollary C).
The Gromoll-Meyer sphere is a 3-sphere-bundle over the 4-sphere.
It is a generator of the group of diffeomorphism classes of oriented homotopy spheres in dimension 7, which is of order 28.
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 construction is due to
Review includes
Generalization of the construction to a large class of exotic 7-spheres:
See also:
Jost-Hinrich Eschenburg, Martin Kerin, Almost positive curvature on the Gromoll-Meyer sphere, Proc. Amer. Math. Soc (arXiv:0711.2987)
Carlos Durán, Thomas Püttmann, A minimal Brieskorn 5-sphere in the Gromoll-Meyer sphere and its applications, Michigan Math. J. Volume 56, Issue 2 (2008), 419-451 (euclid)
Llohann D. Sperança, Pulling back the Gromoll-Meyer construction and models of exotic spheres, Proceedings of the American Mathematical Society 144.7 (2016): 3181-3196 (arXiv:1010.6039)
Llohann D. Sperança, Explicit Constructions over the Exotic 8-sphere (pdf, pdf)
David S. Berman, Martin Cederwall, Tancredi Schettini Gherardini: Curvature of an exotic 7-sphere [arXiv:2410.01909]
For a proof that the Gromoll-Meyer sphere is the only exotic sphere that is a biquotient of a compact Lie group see
Last revised on October 4, 2024 at 05:28:33. See the history of this page for a list of all contributions to it.