Gromoll–Meyer sphere

Key takeaways

Gromoll–Meyer sphere belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Gromoll–Meyer sphere to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Gromoll–Meyer sphere from memory before moving on to harder problems.

Reference excerpt

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 C 5 {\displaystyle \mathbb {C} ^{5}} consider the complex variety (Brieskorn manifold):

a 2 + b 2 + c 2 + d 3 + e 5 = 0. {\displaystyle a^{2}+b^{2}+c^{2}+d^{3}+e^{5}=0.}

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 Sp ⁡ ( 1 ) {\displaystyle \operatorname {Sp} (1)} (isomorphic to SU ⁡ ( 2 ) {\displaystyle \operatorname {SU} (2)} ) acts on the second symplectic group Sp ⁡ ( 2 ) {\displaystyle \operatorname {Sp} (2)} (isomorphic to Spin ⁡ ( 5 ) {\displaystyle \operatorname {Spin} (5)} ) with the embedding Sp ⁡ ( 1 ) ↪ Sp ⁡ ( 2 ) , q ↦ diag ⁡ ( q , q ) {\displaystyle \operatorname {Sp} (1)\hookrightarrow \operatorname {Sp} (2),q\mapsto \operatorname {diag} (q,q)} and multiplication from the left as well as the embedding Sp ⁡ ( 1 ) ↪ Sp ⁡ ( 2 ) , q ↦ diag ⁡ ( q , 1 ) {\displaystyle \operatorname {Sp} (1)\hookrightarrow \operatorname {Sp} (2),q\mapsto \operatorname {diag} (q,1)} and multiplication from the right. A description of the Gromoll–Meyer sphere is the biquotient space:

Sp ⁡ ( 1 ) ∖ Sp ⁡ ( 2 ) / Sp ⁡ ( 1 ) . {\displaystyle \operatorname {Sp} (1)\backslash \operatorname {Sp} (2)/\operatorname {Sp} (1).}

Properties It is the only seven-dimensional exotic sphere which can be expressed as a biquotient of a compact Lie group. It can be expressed as a S 3 {\displaystyle S^{3}} -fiber bundle over S 4 {\displaystyle S^{4}} and hence is a Milnor sphere. Such bundles also include the quaternionic Hopf fibration, whose total space is the ordinary S 7 {\displaystyle S^{7}} . It generates the seventh Kervaire–Milnor group Θ 7 ≅ Z 28 {\displaystyle \Theta _{7}\cong \mathbb {Z} _{28}} .

Literature Gromoll, Detlef; Meyer, Wolfgang (1974). "An Exotic Sphere With Nonnegative Sectional Curvature". Annals of Mathematics. Second Series. 100 (2): 401–406. doi:10.2307/1971078. JSTOR 1971078. Kapovitch, Vitali; Ziller, Wolfgang (2002-10-16). "Biquotients with singly generated rational cohomology". arXiv:math/0210231. Eschenburg, Jost-Hinrich; Kerin, Martin (2007-11-19). "Almost positive curvature on the Gromoll-Meyer sphere". arXiv:0711.2987 [math.DG]. Sperança, Llohann D. (2010-10-28). "Pulling back the Gromoll-Meyer construction and models of exotic spheres". arXiv:1010.6039 [math.DG]. Berman, David S.; Cederwall, Martin; Gherardini, Tancredi Schettini (2025). "Curvature of an exotic 7-sphere". Journal of Geometry and Physics. 216 105590. arXiv:2410.01909. Bibcode:2025JGP...21605590B. doi:10.1016/j.geomphys.2025.105590.

External links Gromoll-Meyer sphere at the nLab

Worked examples

Example 1 — a first encounter with Gromoll–Meyer sphere

Start with the simplest possible case. Write down what Gromoll–Meyer sphere claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Gromoll–Meyer sphere before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Gromoll–Meyer sphere ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Gromoll–Meyer sphere

In research: Gromoll–Meyer sphere appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Gromoll–Meyer sphere in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Gromoll–Meyer sphere is common in secondary-school and first-year university syllabi. It links to neighbouring topics Differential structures, Differential topology, so understanding it makes those chapters shorter.
In everyday life: Look for Gromoll–Meyer sphere outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Gromoll–Meyer sphere in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Gromoll–Meyer sphere means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Gromoll–Meyer sphere out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Gromoll–Meyer sphere in simple terms?

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.

Why does Gromoll–Meyer sphere matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Gromoll–Meyer sphere?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Gromoll–Meyer sphere.