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Gromoll-Meyer sphere

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Idea

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).

Definition

Consider the quaternion unitary group Sp(2) \simeq Spin(5) and its two subgroup inclusion of Sp(1) \simeq SU(2) as

Sp(1){(q 0 0 q)|qSp(1)}AAandAASp(1){(q 0 0 1)|qSp(1)} Sp(1) \;\simeq\; \left\{ \left( \array{ q & 0 \\ 0 & q } \right) \;\vert\; q \in Sp(1) \right\} \phantom{AA} \text{and} \phantom{AA} Sp(1) \;\simeq\; \left\{ \left( \array{ q & 0 \\ 0 & 1 } \right) \;\vert\; q \in Sp(1) \right\}

Then the double coset space/biquotient Sp(1)\Sp(2)/Sp(1)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:

  1. homeomorphic to the topological 7-sphere;

  2. diffeomorphic to an exotic 7-sphere.

As such, it is called the Gromoll-Meyer sphere, due to Gromoll-Meyer 74

Sp(1)\Sp(2)/Sp(1)S exotic 7 Sp(1)\backslash Sp(2) /Sp(1) \;\simeq\; S^7_{exotic}

Properties

The Gromoll-Meyer sphere is the only exotic sphere that can be modeled by a biquotient of a compact Lie group (KZ02, Corollary C).

It is a 3-sphere-bundle over the 4-sphere, and 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 n1 diffSO(n)/SO(n1)S^{n-1} \simeq_{diff} SO(n)/SO(n-1)this Prop.
S 2n1 diffSU(n)/SU(n1)S^{2n-1} \simeq_{diff} SU(n)/SU(n-1)this Prop.
S 4n1 diffSp(n)/Sp(n1)S^{4n-1} \simeq_{diff} Sp(n)/Sp(n-1)this Prop.
exceptional:
S 7 diffSpin(7)/G 2S^7 \simeq_{diff} Spin(7)/G_2Spin(7)/G2 is the 7-sphere
S 7 diffSpin(6)/SU(3)S^7 \simeq_{diff} Spin(6)/SU(3)since Spin(6) \simeq SU(4)
S 7 diffSpin(5)/SU(2)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 diffG 2/SU(3)S^6 \simeq_{diff} G_2/SU(3)G2/SU(3) is the 6-sphere
S 15 diffSpin(9)/Spin(7)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)

References

The construction is due to

  • Detlef Gromoll, Wolfgang Meyer, An Exotic Sphere With Nonnegative Sectional Curvature, Annals of Mathematics Second Series, Vol. 100, No. 2 (Sep., 1974), pp. 401-406 (jstor:1971078)

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)

For a proof that the Gromoll-Meyer sphere is the only exotic sphere that is a biquotient of a compact Lie group see

  • Vitali Kapovitch, Wolfgang Ziller, Biquotients with singly generated rational cohomology, (arXiv:math/0210231)

Last revised on June 18, 2019 at 17:56:59. See the history of this page for a list of all contributions to it.