View source for Gromoll–Meyer sphere

In [[mathematics]], especially [[differential topology]], the '''Gromoll–Meyer sphere''' is a special seven-dimensional [[exotic sphere]] with several unique properties. It is named after [[Detlef Gromoll]] and Wolfgang Meyer, who first described it in detail in 1974, although it was already found by [[John Milnor]] in 1956.

== Definition ==

=== Brieskorn sphere ===
In <math>\mathbb{C}^5</math> consider the [[complex variety]]:

: <math>a^2+b^2+c^2+d^3+e^5=0.</math>

A description of the Gromoll–Meyer sphere is the intersection of the above variety with a small sphere around the origin.

=== Lie group biquotient ===
The first [[symplectic group]] <math>\operatorname{Sp}(1)</math> (isomorphic to <math>\operatorname{SU}(2)</math>) acts on the second symplectic group <math>\operatorname{Sp}(2)</math> (isomorphic to <math>\operatorname{Spin}(5)</math>) with the embedding <math>\operatorname{Sp}(1)\hookrightarrow\operatorname{Sp}(2),
q\mapsto\operatorname{diag}(q,q)</math> and multiplication from the left as well as the embedding <math>\operatorname{Sp}(1)\hookrightarrow\operatorname{Sp}(2),
q\mapsto\operatorname{diag}(q,1)</math> and multiplication from the right. A description of the Gromoll–Meyer sphere is the biquotient space:

: <math>\operatorname{Sp}(1)\backslash\operatorname{Sp}(2)/\operatorname{Sp}(1).</math>

== Properties ==

* It is the only seven-dimensional exotic sphere, which can be expressed as a biquotient of a [[Compact space|compact]] [[Lie group]].

* It can be expressed as a <math>S^3</math>-[[fiber bundle]] over <math>S^4</math> and hence is a [[Milnor sphere]]. Such bundles also include the quaternionic [[Hopf fibration]], whose total space is the ordinary <math>S^7</math>.

* It generates the seventh [[Kervaire–Milnor group]] <math>\Theta_7\cong\mathbb{Z}_{28}</math>.

== Literature ==

* {{cite journal |last=Gromoll |first=Detlef |author-link=Detlef Gromoll |last2=Meyer |first2=Wolfgang |year=1974 |title=An Exotic Sphere With Nonnegative Sectional Curvature |url= |journal=[[Annals of Mathematics]] |series=Second Series |volume=100 |issue=2 |pages=401-406 |jstor=1971078}}
* {{cite arXiv |arxiv=math/0210231 |first=Vitali |last=Kapovitch |first2=Wolfgang |last2=Ziller |title=Biquotients with singly generated rational cohomology |date=2002-10-16}}
* {{cite arXiv |arxiv=0711.2987 |first=Jost-Hinrich |last=Eschenburg |first2=Martin |last2=Kerin |title=Almost positive curvature on the Gromoll-Meyer sphere |date=2007-11-19}}
* {{cite arXiv |arxiv=1010.6039 |first=Llohann D. |last=Sperança |title=Pulling back the Gromoll-Meyer construction and models of exotic spheres |date=2010-10-28}}
* {{cite arXiv |arxiv=2410.01909 |first=David S. |last=Berman |first2=Martin |last2=Cederwall |title=Curvature of an exotic 7-sphere |date=2024-10-02 |first3=Tancredi Schettini |last3=Gherardini}}

== External links ==

* [[nlab:Gromoll-Meyer+sphere|Gromoll-Mayer sphere]] at the [[NLab|''n''Lab]]

[[Category:Differential topology]]