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nLab > Latest Changes: real projective space

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 11th 2017

    added a quick note on the CW-structure on real projective space: here.

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 11th 2017

    Another very common (and slightly different) realization is inductive, via pushouts of type

    S n−1 → D n quot↓ ↓ ℝℙ n−1 ↪ ℝℙ n\array{ S^{n-1} & \to & D^n \\ \mathllap{quot} \downarrow & & \downarrow \\ \mathbb{R P}^{n-1} & \hookrightarrow & \mathbb{R P}^n }

    where quotquot divides by the ℤ/2\mathbb{Z}/2-action.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 11th 2017

    The complex analogue of that happens to be discussed at complex projective space, here.

    • CommentRowNumber4.
    • CommentAuthorSamuel Adrian Antz
    • CommentTimeFeb 4th 2024
    • (edited Feb 4th 2024)

    Added homotopy groups, homology and cohomology for real projective space. Homology has a reference in Hatcher, while homotopy groups and cohomology could still need one to literature (but the former has a reference in Hatcher about the long exxact sequence of homotopy groups of a fiber bundle used to calculate them). Both results can also be found on Topospaces, which is now also linked (just like on complex projective space.

    (I’m not sure, how do the case split correctly? \array{} seems to ignore all spaces inbetween.)

    diff, v8, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeFeb 4th 2024
    • (edited Feb 4th 2024)

    Maybe try the alternative syntax \begin{array}{ll}... \end{array} which allows to enforce alignment just as in actual LaTex.

  6. Unfortnutely, it still doesn’t work out. \begin{cases}… \end{cases} (with the additional feature of generating the large bracket at the beginning automatically) also does not. But I’ve worked out, that \quad still works, although the free space it generates is a bit too large. And of course \leq still gets shown, although < does not (but > does). I’ve now rewritten it in the best form possible with all of this.

  7. Added fixed-point property for real projective space. (Also removed second * in front of Hatcher left over when previously adding it.)

    diff, v9, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeFeb 5th 2024
    • (edited Feb 5th 2024)

    Could you say more explicitly what you feel does not work?

  9. \; and < are not shown inside of the array environment after submitting the page (hence I used \quad and \leq as a replacement). Another example is the section about homology and cohomology on complex projective space (which is not by me).

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeFeb 5th 2024
    • (edited Feb 5th 2024)

    But what are you trying to achieve? Looking at the entry now, the alignment looks correct to me.(?)

    Maybe you could isolate a particular case in the Sandbox and let me know explicitly how you’d like the rendering to be different?

    • CommentRowNumber11.
    • CommentAuthorSamuel Adrian Antz
    • CommentTimeFeb 8th 2024
    • (edited Feb 8th 2024)

    Added topological complexity (for which I intend to create a page in the future) of real projective space and connection to the immersion problem.

    Concering the above rendering problem: I’ve copied some of the formulas into the sandbox. Some of them get shown without spaces inbetween for me (for example „00<kk<nn“ as „0kn0kn“ or „kk odd and 1≤k≤n−11\leq k\leq n-1“ as „kkoddand1≤k≤n−11\leq k\leq n-1“) even with different browsers. Using \qquad instead of \; and \leq instead of < works though.

    diff, v10, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeFeb 8th 2024
    • (edited Feb 8th 2024)

    By the way, there is the “\text{...}“-environment, which should be preferred for saying things like “odd”.

    (That doesn’t address the spacing issue, though, except for whitespace within one text environment.)

  13. Fixed link to topological complexity. Also added BO(1)B O(1) with link to future page.

    diff, v11, current

  14. Added redirects “real projective n-space”, “real projective n-spaces”, “infinite real projective space” and “infinite real projective spaces” like on complex projective space.

    diff, v12, current

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