TITLE: Is there a good way to understand the free loop space of a sphere?
QUESTION [22 upvotes]: I'd like to understand the structure of the free loop space of $S^n$ for small values of $n$.  Here "understand" means roughly that I'd like to know a CW complex with the same homotopy type.
I already "understand" the pointed loop space of $S^n$ using the James reduced product.  (See Hatcher, Algebraic Topology, Section 4.J.)  Is there something analogous for the free loop space?  Alternatively, is there some simple relationship between the free loop space and the pointed loop space?

REPLY [6 votes]: I recently learned the following description of the topology of the loop space at the European Talbot on free loop spaces, and is probably what Ziller does in the paper in the answer of Lennart Meier. 
The idea is to use Morse theory for the standard round sphere. The critical points of the energy functional $E:LS^n\rightarrow \mathbb R$ are the closed geodesics. 
What are the closed geodesics? The trivial ones are the constant loops. The collection of all constant loops forms a family diffeomorphic to $S^n$. Other closed geodesics are great circles. The family of great circles can be described as $T_rS^n$ where $r$ is the energy of a prime great circle. But there are more closed geodesiscs. These are the iterates of the great circles. The iterates form families $T_{k^2r}S^n$, which are of course all diffeomorphic to the unit sphere bundle of $S^n$.
The wonderful luck is that $E$ is Morse-Bott and satisfies Palais-Smale. That means that you can build up $LS^n$ by attaching disc bundles along the critical sets. Thus we see that the homotopy type is
$$
LS^n\simeq S^n\cup D_{\lambda(1)}T_{r}S^n\cup D_{\lambda(2)}T_{2^2r}S^n\cup\ldots
$$
In the union we make some identifications on the boundary of the disc bundles which I do not want to specify here. The number $\lambda(k)$ is the Morse-Bott index of the critical set $T_{k^2r}S^n$ (If I remember correctly it is $(2k-1)(n-1)$). 
One can show that the attachment of the discbundles is homological trivial in this situation. Thus the homology of the loop space the sum of the homology of the sphere and copies of the homology of the unit tangent bundle of the sphere shifted in dimension. 
The unit tangent bundle of the sphere has two torsion if $n$ is even, but not when $n$ is odd. This explains the difference in the homology of the free loop space in the even and odd case. This is Lennart's comment on the Hairy Ball theorem. 
I think that this tactic can be made to work for all manifolds that admit metrics all of whose geodesics are closed (and of the same length).
If one feels like it one can even compute string operations like the Chas-Sullivan product in this manner.