Proof of Real Projective Plane Embedding in R4

[rainwyz0706]

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!

 

[Bacle]

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.

 

[Bacle]

q
S² ---->S²/~
/ |\
f / F
| \/
R⁴


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.?