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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
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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
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Is that correct? I hope that someone who is knowledgeable about this topic can add to the article what the correct attaching map is
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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.
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The attaching map from the boundary of the 8-cell to the 0-cell is trivial, and the resulting space is an 8-sphere.
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The description of the Cayley plane as a CW complex states that it has three cells: one each of dimensions 0, 8, 16.
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non-Desarguesianness of the plane shows that there will be no projective space.
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It seems highly likely that this is the bundle map of the Hopf fibration
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plane. But the open orbit would be the noncompact one.
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