@EricPeterson @ArunDebray Thanks!

Verdier proved the McKay correspondence as a pull-push transform from ADE-orbifolds to their blowups, in equivariant K-theory:

ncatlab.org/nlab/show/McKay+correspondence#GSV85

The same integral transform could be considered with respect to other equivariant spectra.

Has anyone considered the analogous McKay correspondence for other generalized equivariant cohomology theories?

I would like to know what becomes of the statement when one replaces equivariant K-theory with equivariant stable cohomotopy.

@SaalHardali There is also the paper "HP^2 bundles and elliptic cohomology" by Stolz and Kreck. They show that you can obtain real K-theory by taking Spin bordism, quotienting out by the subgroup consisting of classes represented by HP^2 bundles (with map factoring through the base), and inverting the Bott element.

Also what is the precise statement you are after? You certainly can't represent the negative K-homology groups with maps from manifolds. Do you mean up to multiplication by the Bott element?

@ChrisSchommer-Pries I'm not sure about what i mean. Its mainly something that i've seen used in context of index type theorems (AS and GRR etc.). More specifically I was hoping to understand how in Atiyah Singer one can reduce to the case of a dirac operator on a spin manifold. Maybe even the general context in which such a reduction technique is possible.

@SaalHardali See the last section of this paper (which uses the setup from before, in particular the definition of a sort of spin^c suspension of an arbitrary manifold in sec 2.3). I learned this reference from an MO post of Paul Siegel at some point.

But I am guessing this is not really in the direction you want, which seems more philosophical than this.

@MikeMiller I won't describe what i'm looking for as philosphical. For strarters i'd be happy to know the correct geometric description of cycles (as well as the equivalence relation) for both KO and KU homology which has to do with maps from spin manifolds. I don't know the precise statement but its along the lines of "any cycle can be represented as map from a such and such manifold equipped with such and such vector bundle upto such and such equivalence relation"

Yeah, I realized what I should have said was "less computational".

Plus the added piece of information that I think that for KU one should take Spin^c and for KO, Spin.

I don't know if this is obvious for everyone already (or wrong, for that matter), but is it possible that the result was about ko and ku, rather than KO and KU? After all, the ABS map MSpin -> KO factors through ko -> KO (and the same for MSpinc and KU).