Homework 3 Problem 2

Problem

The real projective plane $\mathbb{RP}^2$ is obtained by identifying opposite edges of a rectangle with one another as shown.

What is the smallest integer $m$ for which $\mathbb{RP}^2$ can be immersed in $\mathbb{R}^m$? What is the smallest integer $n$ for which $\mathbb{RP}^2$ can be embedded in $\mathbb{R}^n$? Intuitive responses are sufficient if they're explained convincingly.

Solution

The smallest integer $m$ for which $\mathbb{RP}^2$ can be immersed in $\mathbb{R}^m$ is 3. The reasoning for this follows from the discussion we had in class about the torus and the Klein bottle. A rectangular piece of paper could be folded in order to match opposite edges with one another as shown, provided the paper allows for self-intersection. The fact that $\mathbb{RP}^2$ self-intersects in $\mathbb{R}^3$ disqualifies this from being an embedding.

The smallest integer $n$ for which $\mathbb{RP}^2$ can be embedded in $\mathbb{R}^n$ is 4. Since it is immersed in $\mathbb{R}^3$, $\mathbb{RP}^2$ can be represented by homogeneous coordinates $(x,y,z) \in \mathbb{R}^3$ in the equation $x^2 + y^2 + z^2 = 1$ subject to the relation that $(x,y,z)=(-x,-y,-z)$. The embedding is then given by the function $(x,y,z) \mapsto (xy,xz,y^2-z^2,2yz)$. This mapping of $\mathbb{RP}^2$ into $\mathbb{R}^4$ does not result in any self-intersections, thus qualifying this to be an embedding.

Discussion

Right?

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