## Projective Space Explicit Homotopy (RP1 to RP2)

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The above video describes the real projective plane ($\mathbb{R}P^2$).

The projective space $\mathbb{R}P^n$ can be defined as the quotient space of $S^n$ by the equivalence relation $x\sim -x$ for $x\in S^n$.

Notation: For $x=(x_1,\dots, x_{n+1})\in S^n$, we write $[x_1, x_2,\dots, x_{n+1}]$ for the corresponding point in $\mathbb{R}P^n$. Let $f,g: \mathbb{R}P^1\to\mathbb{R}P^2$ be the maps defined by $f[x,y]=[x,y,0]$ and $g[x,y]=[x,-y,0]$.

How do we construct an explicit homotopy between $f$ and $g$? A common mistake is to try  the “straight-homotopy”, e.g. $F([x,y],t)=[x,(1-2t)y,0]$. This is a mistake as it passes through the point [0,0,0] which is not part of the projective plane.

A better approach is to consider $F:\mathbb{R}P^1\times I\to\mathbb{R}P^2$, defined by $\boxed{F([x,y],t)=[x,(\cos\pi t)y, (\sin\pi t)y]}$.

Note that if $x^2+y^2=1$, then $x^2+[(\cos\pi t)y]^2+[(\sin\pi t)y]^2=x^2+y^2=1$.

$F([x,y],0)=[x,y,0]$

$F([x,y],1)=[x,-y,0]$