11:03 PM
Draw the straightline homotopy between the antipodal and the identity map S^2 -> S^2 $\subset$ R^3 at time 0, 1/4, 1/2, 3/4, 1
Send the pictures over to me
I mean, I guess a simple enough homotopy is just contracting everything to 0 and then expanding back but antipodally this time but this loses all info about the map
Straightline homotopy does that when crossing 0
So that's also another thing
You shouldn't collapse substantial parts of your space in the process of modifying your map to something understandable
I guess what I described is exactly the straightline homotopy. so yeah it's useless for a different reason
But you can imagine drawing the time-slices for a straightline homotopy be terrible for a general map $M \to M$ of manifolds, where you Whitney embed $M$ in $\Bbb R^n$