Question

Show that on any surface, homotopic simple closed curves are
isotopic.

Answer 1

For any surfaces there are many statements along the lines of: if two simple closed curves are homotopic, they are isotopic.such questions for families of curves.

More precisely,

let Σ be a hyperbolic surface, possibly with boundary. We fix an essential simple closed curve ?on Σ. It is true that the subspace of ???(s¹,Σ)Emb(s¹,Σ) consisting of those curves that are isotopic to ?is homotopy equivalent to a circle.

Here the circle would come from reparametrisation of the curves.

This statement is true if we instead look at the space of all continuous (or smooth) maps of ?1 into Σ that are homotopic to ?.

Also note that this seems to be false for the torus, as for any essential simple closed curve we get at least ?¹×?¹