CW complex of iterated loop spaces In Milnor's book Morse Theory, it is proved that  the loop space $\Omega S^n$ of the n sphere has the homotopy type of a CW complex with one cell each in the dimensions 0, n-1, 2n-2, 3n-3, ... Or more generally, given non conjugate points p, q on a complete Riemannian Manifold M, the path space $\Omega(M,p,q)$ (of all continuous path joining p to q) has the homotopy type of a countable CW complex which contains one cell of dimension $d$ for each geodesic from p to q of Morse index $d$.

For iterated loop spaces $\Omega^k M$, do we have a similar theorem? Say is there any known result concerning the CW structure of $\Omega^k S^n$?

 A: By a result of Milnor, the space of maps from a finite CW complex to any CW complex is homotopy equivalent to a CW complex. This gives a general reason why spaces like $\Omega^k M$ have a CW structure.
There is a well-known construction from which you can, at least in principle, get an explicit cell structure for spaces of the form $\Omega^k \Sigma^k X$, where $X$ is a based path-connected CW-complex. This includes $\Omega^k S^n$ as a special case (for $n>k$). The general construction goes as follows. For a finite set $i$, let $F(\mathbb R^k; i)$ be the space of injective maps from $i$ to $\mathbb R^k$. The assignment $i\mapsto F(\mathbb R^k; i)$ gives a contravariant functor from the category of finite sets and injections to the category of topological spaces. Similarly we have a covariant functor between same categories, $i\mapsto X^i$, where the functor structure is given by basepoint-inclusions. Given a pair of functors like this, one may form the coend $$F(\mathbb R^k; i)\otimes_i X^i.$$ A classic theorem of (I think) Milgram, May and Segal asserts that this coend is homotopy-equivalent to $\Omega^k\Sigma^k X$. This endows the homotopy type of $\Omega^k\Sigma^k X$ with a natural filtration, and it also can be used to equip it with a CW structure, since the spaces $F(\mathbb R^k; i)$ can be endowed with CW structures compatible with maps between them. One can get different CW models for the homotopy type of $\Omega^k\Sigma^k X$ by using cellular approximations of the functor $i\mapsto F(\mathbb R^k; i)$. Specific cellular approximations were constructed by Milgram, Barratt-Eccles, Jeff Smith, and possibly some others. This paper of Clement Berger gives a nice historical survey of constructions of this type.
When $k=1$ there is a particularly small CW model, as you indicated. In other cases it is not going to be so simple to enumerate the cells. Nevertheless, this construction is useful for many purposes. For example, it was used to describe the homology of $\Omega^k\Sigma^k X$ (with field coefficients) as a functor of the homology of $X$.
