I am curious about the topology of the space of simple closed curves in $S^2$.

The entire free loop space $LS^2$ admits an explicit description using the Morse theory of the energy functional for the round metric. The energy functional is Morse-Bott, with a sequence of critical manifolds corresponding to iterates of great circles. Topologically, this implies that $LS^2$ is homotopy equivalent to the complex constructed by starting with $S^2$ (the space of constant loops) and successively attaching copies of the unit disk tangent bundle of $S^2$ in some manner.

Given this context, my question can be phrased as follows. Let $\mathcal{S}S^2 \subset LS^2$ be the space of simple closed curves, i.e. injective continuous maps $S^1 \to S^2$. Does $\mathcal{S}S^2$ admit a similar nice description?

If the gradient flowline of the energy functional starting at any simple closed curve ended at a non-multiply-covered great circle, then it seems to me that this would give a retraction of $\mathcal{S}S^2$ onto the unit disk tangent bundle to $S^2$. However, I don't actually know if this is true or not.

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    $\begingroup$ Use the fiber sequence $\text{Diff}^+(S^2, S^1) \to \text{Diff}^+(S^2) \to \text{Emb}(S^1, S^2)$, where the first group is oriented homeomorphisms that fix the equator pointwise. Some work with Smale's theorem on diffeomorphisms of the disc show that the first space is contractible and the second deformation retracts to $SO(3)$, so the space of embeddings deformation retracts to great circles. Not an answer because I don't quite follow what your subspace is. $\endgroup$
    – mme
    Dec 13, 2020 at 22:53
  • $\begingroup$ Hi Mike, I think this answers my question, I was indeed looking for the homotopy type of $\text{Emb}(S^1, S^2)$. I'll reword the question to make it more clear shortly. $\endgroup$ Dec 13, 2020 at 23:00

1 Answer 1


Yes; I find this easier than the whole Morse theory package needed for $LS^2$.

Thm. The projection map $\text{Emb}(S^1, S^2) \to (TS^2 \setminus 0)$, given by $\gamma \mapsto \gamma'(0)$, is a weak homotopy equivalence.

A section sends a nonzero tangent vector $v$ above a point $p$ to the geodesic through $p$ with $\gamma'(0) = v$; it should be the case that $\text{Emb}(S^1, S^2)$ deformation retracts to the space of great circles (which is homeomorphic to $SO(3)$); I don't know a reference to the claim that the corresponding inclusion is a cofibration so that I can conclude. (Maybe there is an explicit deformation retraction from mean curvature flow.)

Pf. First observe that $\text{Diff}^+(S^2)$ acts transitively on $\text{Emb}(S^1, S^2)$ by the fact that the latter space is path-connected (roughly, the Schoenflies theorem) and the isotopy extension theorem. This (plus some general theorems about Frechet group actions) show that there is a fiber sequence $$\text{Diff}^+(S^2 \text{ rel } S^1) \to \text{Diff}^+(S^2) \to \text{Emb}(S^1, S^2).$$ The first space is the space of oriented diffeomorphisms which fix the equator pointwise. The same derivative map above defines a map of fiber sequences to $\{*\} \to TS^2 \setminus 0 \to TS^2 \setminus 0$ (sending a diffeomorphism $\varphi$ to $d\varphi_p(e_1)$, where $p$ is a point on the equator and $S^1$ the tangent vector corresponding to the equator.

The claim is that this the first space is contractible and the map $\text{Diff}^+(S^2) \to TS^2 \setminus 0$ is a weak equivalence.

Both of these claims follow from Smale's theorem $\text{Diff}(D^2 \text{ rel } \text{Nbhd}(\partial D^2)) \simeq \{*\}$ with a little massaging, where here my notation means "the space of diffeomorphisms of the disc which fix a neighborhood of the boundary".

First you show that $\text{Diff}(S^2 \text{ rel } S^1)$ deformation retracts onto the space of diffeomorphisms which are linear on an open nbhd of $S^1$ in a fixed trivialization of a tubular neighborhood of $S^1$, then use the fact that the space of oriented linear isomorphisms $NS^1 \to NS^1$ is contractible; this buys you that $\text{Diff}^+(S^2 \text{ rel } S^1)$ is equivalent to the space of diffeomorphisms which are constant on a neighborhood of the equator; now Smale's result applies.

For $\text{Diff}^+(S^2)$ itself, the argument is similar. The fiber of the map described above is the set of diffeomorphisms fixing a point and a tangent vector at that point; because the map is orientation-preserving, this leaves a contractible space of possible derivatives, and you can show this fiber deformation retracts onto the space of diffeomorphisms fixing a chosen point and with $d\varphi_x = \text{Id}$; then linearize, as above, so that this fiber is equivalent to the space of diffeomorphisms of $S^2$ which fix an open neighborhood of $p$ pointwise, which again is contractible by Smale.


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