Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 $S^1 \times S^1$.

share|cite|improve this question

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

share|cite|improve this answer
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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.