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nLab > Latest Changes: Arnold-Kuiper-Massey theorem

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeAug 4th 2020
- (edited Aug 4th 2020)
- PermaLink
Author: Urs
Format: MarkdownItexthis used to be inside _[[4-sphere]]_. Am giving it its stand-alone entry for ease of listing references <a href="https://ncatlab.org/nlab/revision/Arnold-Kuiper-Massey+theorem/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Arnold-Kuiper-Massey+theorem">current</a>

this used to be inside 4-sphere. Am giving it its stand-alone entry for ease of listing references

v1, current
- CommentRowNumber2.
- CommentAuthorUrs
- CommentTimeFeb 7th 2021
- PermaLink
Author: Urs
Format: MarkdownItexI have added statement of and references for the generalizations of the theorem to quaternionic- and octonionic projective planes: $$ \mathbb{H}P^2 / \mathrm{U}(1) \simeq S^7 $$ $$ \mathbb{O}P^2 / \mathrm{Sp}(1) \simeq S^{13} $$ <a href="https://ncatlab.org/nlab/revision/diff/Arnold-Kuiper-Massey+theorem/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/Arnold-Kuiper-Massey+theorem/4">v4</a>, <a href="https://ncatlab.org/nlab/show/Arnold-Kuiper-Massey+theorem">current</a>

I have added statement of and references for the generalizations of the theorem to quaternionic- and octonionic projective planes:
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>ℍ</mi><msup><mi>P</mi> <mn>2</mn></msup><mo stretchy="false">/</mo><mi mathvariant="normal">U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>S</mi> <mn>7</mn></msup></mrow><annotation encoding="application/x-tex"> \mathbb{H}P^2 / \mathrm{U}(1) \simeq S^7 </annotation></semantics></math>$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>𝕆</mi><msup><mi>P</mi> <mn>2</mn></msup><mo stretchy="false">/</mo><mi mathvariant="normal">Sp</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>S</mi> <mn>13</mn></msup></mrow><annotation encoding="application/x-tex"> \mathbb{O}P^2 / \mathrm{Sp}(1) \simeq S^{13} </annotation></semantics></math>$
diff, v4, current
- CommentRowNumber3.
- CommentAuthorDavid_Corfield
- CommentTimeNov 13th 2021
- (edited Nov 13th 2021)
- PermaLink
Author: David_Corfield
Format: MarkdownItexI see Arnold in his [article](https://ncatlab.org/nlab/show/Arnold-Kuiper-Massey+theorem#Arnold99) sets the $S^4$ result in a different series, in terms of a trinity $$\mathbb{R}P^1 \simeq S^1; \mathbb{C}P^2/Conj \simeq S^4; (\mathbb{H}P^4/Aut(\mathbb{H}))/Conj \simeq S^13.$$

I see Arnold in his article sets the $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">S^4</annotation></semantics></math>$ result in a different series, in terms of a trinity
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>ℝ</mi><msup><mi>P</mi> <mn>1</mn></msup><mo>≃</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>;</mo><mi>ℂ</mi><msup><mi>P</mi> <mn>2</mn></msup><mo stretchy="false">/</mo><mi>Conj</mi><mo>≃</mo><msup><mi>S</mi> <mn>4</mn></msup><mo>;</mo><mo stretchy="false">(</mo><mi>ℍ</mi><msup><mi>P</mi> <mn>4</mn></msup><mo stretchy="false">/</mo><mi>Aut</mi><mo stretchy="false">(</mo><mi>ℍ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>Conj</mi><mo>≃</mo><msup><mi>S</mi> <mn>13</mn></msup><mo>.</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}P^1 \simeq S^1; \mathbb{C}P^2/Conj \simeq S^4; (\mathbb{H}P^4/Aut(\mathbb{H}))/Conj \simeq S^13.</annotation></semantics></math>$

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nLab > Latest Changes: Arnold-Kuiper-Massey theorem