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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 $S^1 \times S^1$.

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2 Answers 2

up vote 3 down vote accepted

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

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I was just looking at this Gramain reference and it seems only to prove something which is equivalent, though perhaps not obviously so. Namely, Theorem 4 of the paper says the space of embeddings of a circle with a fixed 1-jet at a point has contractible components. This is the fiber of a map from the full embedding space which takes the 1-jet at a point. The base space of this fibration is essentially the unit tangent bundle of the surface. There's a little checking left to get the result from this, but it's not hard. Does anyone know a reference for the exact statement? –  Allen Hatcher Aug 21 '12 at 23:13

If you forget about the parametrization, the "curve shortening flow" isotopes an essential simple closed curve to THE geodesic isotopic to it (this is a celebrated result of Matt Grayson), which I believe is gives a deformation retraction of the unparametrized space to a point. When you throw the parametrization back in, you get your conjectured result.

The Grayson result is this: Shortening embedded curves MA Grayson - The Annals of Mathematics, 1989

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