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.




About mathtuition88
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