Let $M$ be a compact manifold and let $f$ be a Morse function with exactly one critical point at each critical level. Then one can recover a CW-complex with the homotopy type of $M$ from just the critical point data of $f$: simply go through the critical points of $f$ in order and attach an $n$-cell every time you pass through a critical point of index $n$.

Hence it is possible to recover any homotopy theoretic information you could want from $M$. In some cases it is possible to directly calculate topological invariants using Morse theory; for instance, one can recover the cohomology ring of $M$ using Morse cohomology.

I'm wondering if there is a tool in the spirit of Morse cohomology which recovers the topological K-theory of $M$. I'm looking for something more effective than "use a Morse function to build a CW-complex and calculate its K-theory." For instance, there should be a natural Chern character map from "Morse K-theory" to Morse cohomology.

I haven't seen anything like this in the literature, but that certainly doesn't mean it isn't there. Any ideas?