Space of simple closed curves in $S^2$

Question

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.

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.