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Real projective plane

  1. Apr 23, 2010 #1
    I'm working on a proof to show there exists an embedding of the real projective plane P R2 in R4.
    The initial setup is as follows:
    Let S2 denote the unit sphere in R3 given by S2 = {(x, y, z) ∈ R3 : x2 + y 2 + z 2 = 1}, and let
    f : S2 → R4 be defined by f (x, y, z) = (x2 − y 2 , xy, yz, zx).
    I'm trying to show that f determines a continuous map F: P R2 → R4 where P R2 is the real projective plane,
    then show that F is a homeomorphism onto a topological subspace of R4 .
    I think it's easy to see that f(x1,y1,z1)=f(x2,y2,z2) implies (x1,y1,z1)=+/-(x2,y2,z2). But I don't know how to figure out the whole proof completely. Could anyone please give me a hint? Any input is appreciated!
     
  2. jcsd
  3. Apr 24, 2010 #2
    Hi, Rain:

    If a map is constant ( has a constant value) on equivalence classes, then we

    say that the map passes to the quotient, i.e., that we can work out a commutative

    diagram with one of the sets being S/~ , where ~ is the equivalence relation.

    In your case, ~ is the quotient map defined on S^2 . Show that all (2) elements

    in each equivalence class are sent to the same value, so that the other map

    passes to the quotient.
     
  4. May 5, 2010 #3
    q
    S<sup>2</sup> ---->S<sup>2</sup>/~
    / |\
    f / F
    | \/
    R<sup>4</sup>


    Then the map you want is the only map that will make the above diagram commute
    (double check everything, since I am kind of tired now.), a map

    You can then use properties of the quotient topology, and the fact that a
    continuous bijection between compact and Hausdorff is a homeomorphism,
    to show the map is an embedding.


    Does that Answer your Question.?
     
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