Proof of Real Projective Plane Embedding in R4

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In summary, the conversation discusses the proof of the existence of an embedding of the real projective plane in R4. The setup includes the unit sphere in R3 and a map f defined on S2. The goal is to show that f determines a continuous map F from the real projective plane to R4, which is a homeomorphism onto a topological subspace of R4. The conversation also mentions the use of equivalence classes and properties of the quotient topology in the proof. A hint is given to show that the map passes to the quotient and to use the properties of a continuous bijection between compact and Hausdorff spaces to show that the map is an embedding.
  • #1
rainwyz0706
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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!
 
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  • #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.
 
  • #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.?
 

What is a real projective plane embedding in R4?

A real projective plane embedding in R4 refers to the mathematical concept of representing a real projective plane, which is a non-orientable surface, in four-dimensional Euclidean space. This embedding allows for the study and visualization of the projective plane in a more familiar and accessible space.

How is a proof of real projective plane embedding in R4 established?

A proof of real projective plane embedding in R4 involves demonstrating that the properties and characteristics of the projective plane can be accurately represented in four-dimensional space. This is typically done through the use of mathematical equations and transformations.

What is the significance of proving the real projective plane embedding in R4?

Proving the real projective plane embedding in R4 is significant because it allows for a better understanding and visualization of the projective plane, which has important applications in various fields of mathematics and physics. It also adds to the body of knowledge and understanding of higher-dimensional spaces.

Are there any challenges in proving the real projective plane embedding in R4?

Yes, there are several challenges in proving the real projective plane embedding in R4. This includes the complexity of higher-dimensional spaces, the need for advanced mathematical techniques, and the potential for errors or inconsistencies in the proof. Additionally, there may be differing opinions and approaches to the proof, leading to debates and further research.

How does the real projective plane embedding in R4 differ from other mathematical embeddings?

The real projective plane embedding in R4 is unique in that it involves representing a non-orientable surface in a higher-dimensional space. Other mathematical embeddings may involve different types of surfaces or lower-dimensional spaces. Additionally, the real projective plane embedding has important applications in fields such as topology and differential geometry, making it a particularly significant and interesting concept.

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