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nLab > Latest Changes: octonionic projective space

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 27th 2020
    • (edited Nov 27th 2020)

    starting something, for the moment just so as to record that

    there is a homeomorphism

    𝕆P 2≃S 15∪h 𝕆ℍP 1 \mathbb{O}P^2 \,\simeq\, S^{15} \underset{h_{\mathbb{O}}}{\cup} \mathbb{H}P_1

    between the octonionic projective plane and the attaching space obtained from the octonionic projective line along the octonionic Hopf fibration.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 27th 2020

    finally realized that we already have Cayley plane. Have added cross-links now.

    diff, v4, current

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 28th 2020

    Hmm, is that meant to be 𝕆P 1\mathbb{O}P^1?

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 28th 2020

    Yes, I changed it already on the page itself.

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeMar 28th 2021
    • Rowena Held, Iva Stavrov, Brian VanKoten, (Semi-)Riemannian geometry of (para-)octonionic projective planes, Diff. Geom. & its Appl. 27:4 (2009) 464-481 doi:/10.1016/j.difgeo.2009.01.007

    diff, v7, current

  6. Added homotopy groups and cohomology of octonionic projective plane, already stated in Lackman 19 (https://arxiv.org/abs/1909.07047) already present on the page, and also in Mimura 67 (https://doi.org/10.1215/kjm/1250524375) now added as a new reference as well. Replaced 15-sphere by 16-disk in Proposition 2.2. as stated by both of those sources. (The same edit was done on Cayley plane.)

    diff, v8, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeFeb 3rd 2024

    Thanks for the addition and for fixing that bad typo!

    I have added publication data for Lackmann’s article.

    diff, v9, current

    • CommentRowNumber8.
    • CommentAuthorperezl.alonso
    • CommentTimeFeb 3rd 2024

    Something that I tried to highlight here is that these algebras coming from the Cayley-Dickson process are perhaps better understood not as “algebras lacking properties” but as “algebras with weakened properties”. Would it be possible then that the higher dimensional octonionic spaces do not have a manifold description but do admit a stacky presentation?

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeFeb 4th 2024

    Hi Alonso, we had talked about this in another thread (here): Is it actually the case?

    It’d be interesting if it were, but do you have an actual example?

    • CommentRowNumber10.
    • CommentAuthorperezl.alonso
    • CommentTimeFeb 4th 2024

    I’m hoping to look into this when I get some time, but still wanted to pose the question in case there was already some work in this direction (the stacky character of projective spaces).

1 to 10 of 10

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