What is known about K-theory and K-homology groups of (free) loop spaces?

Question

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?

Answer 1

There are a lot of computational methodologies from algebraic topology that you can apply here, moving from less to more complicated. Suppose E_* and E^* is a pair of a generalized homology theory and its cohomology theory, which has a commutative and associative product, and you have a space X where you are interested in the loop space ΩX and the free loop space LX, which live in fibration sequences ΩX -> PX -> X and ΩX -> LX -> X. (Here PX is contractible.)

There are Atiyah-Hirzebruch spectral sequences

Hp(Y; Eq(*)) => Ep+q(Y)
Hp(Y; Eq(*)) => Ep+q(Y)

which, because they are generic, do not have such stellar behavior except in "easy" cases.

If you have a good grip on the (co)homology of X, then there are the Serre spectral sequences associated to the path-loop fibration

Hp(X; Eq(ΩX)) => Ep+q(*)
Hp(X; Eq(ΩX)) => Ep+q(*)

and those associated to the free-loop fibration

Hp(X; Eq(ΩX)) => Ep+q(LX)
Hp(X; Eq(ΩX)) => Ep+q(LX)

The Serre spectral sequence is sometimes less-than-spectacular for loop spaces and free loop spaces, again because it's pretty generic, and because to use then to compute for the loop space you have to play the fiber off the base. This leads to nasty inductive arguments.

Then there are the Eilenberg-Moore spectral sequences for ΩX. If E^*X is a flat E^*-module and X is simply connected, then you get a spectral sequence

Tor**E*X(E*,E*) => E*ΩX

where this is Tor of graded modules over a graded algebra and inherits a bigrading. This is usually much more straightforward than the standard technique of playing the Serre spectral sequence game to find the homology of the fiber. There's also a homology version but it involves CoTor for comodules over E_*.

There's also an Eilenberg-Moore spectral sequence starting with Tor over the cohomology of X of the cohomology of LX with the ground ring, and converging to the cohomology of ΩX. This is often less useful because usually you want to go the opposite direction, but it exists.

Finally, there is the Hochschild homology spectral sequence for LX. If E^*X is a flat E^*-module and X is simply connected, then there is a spectral sequence

HHE*(E*X,E* X) => E*LX

where this is Hochschild homology of E^*X (over the ground ring E^*) with coefficients in itself. This is a graded algebra over a graded ring and the Hochschild homology recovers a bigrading. If you instead took coefficients in the ground ring E^* you're recover the Eilenberg-Moore spectral sequence for the based loop space ΩX.

For example, if E^* X is a polynomial algebra over E^* on classes in even degree, the cohomology of the loop space is exterior and the cohomology of the free loop space is the de Rham complex. More complicated cohomology yields more complicated behavior.

If you have specific spaces in mind then there are more specialized results. For example, one major theorem is the Atiyah-Segal theorem relating the K-theory of the classifying space of a compact Lie group to a completion of its complex representation ring. This is very hard to extract from the above general methods.

(Somebody who is an expert in string topology should step in and talk about K-theory of free loop spaces!)

Answer 2

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.