TITLE: What is known about K-theory and K-homology groups of (free) loop spaces?
QUESTION [15 upvotes]: Calculating the homology of the loop space and the free loop space is reasonably doable. There exists the Serre spectral sequence linking the homology of the loop space and the homology of the free loop space. Furthermore, for finite CW complexes the James product construct a homotopy approximation to the loop space of a suspension.
One wonders how to extend these calculations to generalized (co)homology theories, like K-theory and K-homology. For which spaces do we know the K-theory and/or K-homology groups? How can one calculate these?

REPLY [6 votes]: I take the liberty to reference to a paper of mine (which is hopefully no bad style): Spectral Sequences in String Topology. Here, the K-homology of the free loop spaces of spheres and complex projective spaces is computed (up to extensions when torsion occurs). The method is actually pretty straightforward: one first finds concrete manifold generators for the homology of the free loop spaces of these spaces, which one can show to have a (stably) almost complex structure. This shows that the Atiyah-Hirzebruch spectral sequence for $MU$ degenerates and K-homology is determined by complex cobordism.
Additionaly, I want to comment that calculating the ordinary homology of free loop spaces is in general not that easy, too. For example, in the paper of Cohen, Jones and Yan, where the compute the Chas-Sullivan product for spheres and projective spaces, they reference for the computation of the additive homology of free loop spaces of projective spaces to a paper by Ziller, where he uses a Morse homology method, which, at least to my mind, is not altogether easy. Indeed, there are very few computations of homologies of free loop spaces, as far as I know. Spectral Sequences are a nice tool, but if they don't collapse calculations tend to be not straightforward.