# nLab Gromoll-Meyer sphere

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

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

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

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

Then the double coset space/biquotient $Sp(1)\backslash Sp(2) /Sp(1)$ with respect to the left and right multiplication action, respectively, of these two subgroups is, with respect to its canonically induced geometric structures:

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

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

## Properties

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 (from FSS 19, 3.4)

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)