TITLE: Is there an effective way to calculate K-theory using Morse functions?
QUESTION [16 upvotes]: 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?

REPLY [2 votes]: Roland Voigt has submitted his thesis "Transport functions and Morse K-theory" at Universität Leipzig -- his advisor was Matthias Schwarz. He has not published his results so far and the thesis will only become available (automatically) once he has defended.
You can contact him, his email address is (Roland.Voigt@math.uni-leipzig.de).