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?



Is there a good way to understand the free loop space of a sphere?
Stably the free loop space of the suspension of a connected space splits up, just as the based loop space does. Just as $\Omega\Sigma X$ is stably the wedge of the smash products $X^{\wedge n}$, $L\Sigma X$ is stably the wedge of $S^1\_+\wedge\_{C\_n}X^{\wedge n}$. Here $C\_n$ is a cyclic group of order $n$ acting freely on $S^1$ and permuting the factors in the smash product. This is even correct $S^1$-equivariantly in a weak sense.


EDIT Here is an answer more directly relevant to the question: 


Suppose $X\sim BG$. Even if the group $G$ is not discrete, we still have the equivalence that Qiaochu mentioned in a comment, between $LX$ and the homotopy orbit space for the conjugation action of $G$ on $G$. And this in turn is equivalent to the two-sided bar construction for $G$ acting on itself on both sides, i.e. the cyclic bar construction $N^{cyc}G$. And if we have another topological monoid $M$ equivalent to $G$, for example the Moore loops on $X$, or the James construction $JY$ if $X=\Sigma Y$, then $N^{cyc}M$ is another model for $LX$.


Furthermore, there is a nice way of thinking of (the realization of the simplicial space) $N^{cyc}M$, which informally goes like this: A point is given by a finite subset $T$ of $S^1$ together with a labeling: a function $T\to M$. The set $T$ and the labels can move. If $T$ moves in such a way that several points come together, the label of the new point is the product of the old labels. If a label becomes $1\in M$ then the point may be deleted from $T$.


This is even equivariantly correct regarding the $S^1$-action, in some sense (not good for fixed-point spaces of the whole group, but OK for finite subgroups).


When $X=S^{n}$ and $M=JS^{n-1}$, this gives a pretty good equivariant cell structure for $LS^n$.


