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nLab > Latest Changes: cell structure of projective spaces

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 24th 2020

    am splitting this off as a stand-alone statement (from complex projective space)

    Have cleaned-up the formulation of statement and proof and have generalized from ground ring the complex numbers to reals, complex numbers and quaternions.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJan 29th 2021

    have re-arranged the flow of the proof somewhat. (Previously, half of the proof idea was crammed into the statement of the proposition…)

    I really came back here to make a note on the compatibility of cell attachments as we pass from ℂ\mathbb{C} to ℍ\mathbb{H}:

    S 4k+3 ⟶ D 4(k+1) ↓ (po) ↓ ℂP 2k+1 ⟶ ℂP 2k+2 ↓ (po) ↓ ℍP k ⟶ ℍP k+1 \array{ S^{4k+3} &\longrightarrow& D^{4(k+1)} \\ \big\downarrow &{}^{{}_{(po)}}& \big\downarrow \\ \mathbb{C}P^{2k+1} &\longrightarrow& \mathbb{C}P^{2k+2} \\ \big\downarrow &{}^{{}_{(po)}}& \big\downarrow \\ \mathbb{H}P^{k} &\longrightarrow& \mathbb{H}P^{k+1} }

    But now I am too tired. Maybe tomorrow…

    diff, v6, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJan 30th 2021

    So now I wrote out (here) statement and proof of the above pasting composite.

    (This is all elementary and/or trivial. I am just writing it out for peace of mind. Just recently, in another thread, we had a mistake in a similarly “obvious” homotopy-pasting composite, so it’s good to double-check.)

    diff, v8, current

1 to 3 of 3

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