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


More precisely, let $\Sigma$ be a hyperbolic surface, possibly with boundary. We fix an essential simple closed curve $\gamma$ on $\Sigma$. It is true that the subspace of $Emb(S^1,\Sigma)$ consisting of those curves that are isotopic to $\gamma$ 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 $S^1$ into $\Sigma$ that are homotopic to $\gamma$. Also note that this seems to be false for the torus, as for any essential simple closed curve we get at least $S^1 \times S^1$.



What is the homotopy type of the space of simple closed curves isotopic to a given one?
Earlier than Grayson, the determination of the homotopy-types of these spaces was done by Gramain. 


There are a few special cases, like the torus and sphere and the non-orientable analogue, the case of null curves. But if they're not null homotopic the components of the embedding space have the homotopy type of $S^1$ -- the reparametrizations of the given curve. 


Andre Gramain, Le type d'homotopie du groupe des diffeomorphisms d'une surface compacte. Ann. Sci. l'ENS $4^e$ serie tome 6 $n^o$ 1 (1973) 53--66


