projective space – Mathtuition88

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.