Talk:Cayley plane
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Contradiction in definition
[edit]I'm a mathematician but not an expert on the Cayley plane. That said, I could see a contradiction in the descriptions. The real Cayley plane is given as an orbit of a compact group, hence is compact. The complex Cayley plane has two orbits, and the page says that the open one is the real Cayley plane. But the open orbit would be the noncompact one.
Probably this page should also include the fact that compact F_4 acts by isometries, whereas complex E_6 acts only by collineations. And definitely the statements that one can't use the standard construction of projective planes because it involves dividing by a group, plus the fact that the non-Desarguesianness of the plane shows that there will be no projective space. — Preceding unsigned comment added by 68.175.139.154 (talk) 11:09, 24 May 2016 (UTC)
CW complex description
[edit]The description of the Cayley plane as a CW complex states that it has three cells: one each of dimensions 0, 8, 16.
The attaching map from the boundary of the 8-cell to the 0-cell is trivial, and the resulting space is an 8-sphere.
But the only omitted information is what the attaching map is from the boundary of the 16-cell (a 15-sphere) to the 8-sphere.
It seems highly likely that this is the bundle map of the Hopf fibration S15 → S8.
Is that correct? I hope that someone who is knowledgeable about this topic can add to the article what the correct attaching map is S15 → S8. 2601:200:C082:2EA0:84E9:4DC1:61A8:8BC4 (talk) 19:22, 4 February 2023 (UTC)