Is there an effective way to calculate K-theory using Morse functions? 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?
 A: 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).
A: This really is not an answer, but its longer than a comment  and that is why I write it here. 
$\newcommand{\bR}{\mathbb{R}}$   Suppose that $M$ is a smooth compact manifold of dimension $m$ and $f: M\to \bR$ is a  self-indexing Morse function with critical values 
$$ 0< 1<\cdots <m. $$
We get an increasing filtration  by closed sets
$$ M_0=\emptyset,\;\; M_1=\Bigl\{f\leq \frac{1}{2}\Bigr\},\dotsc, M_m=\Bigl\{ f\leq m-\frac{1}{2}\Bigr\},\;\;M_{m+1}=M. $$
Associated to this filtration  is a a homological spectral sequence. The self-indexing assumption guarantees that  this homological spectral sequence degenerates at $E_1$. The  complex $E_1$. Onse with fix a metric such that the resulting gradient flow satisfies Smale's transversality conditions, the the $E_1$ complex  can  be identified with the homological Morse-Floer complex; see p.62-63 of these notes.
This  filtration also leads to a $K$-theoretic spectral sequence, the Atiyah-Hirzebruch spectral sequence. This should be viewed as a first approximation for a $K$-theoretic    Morse-Floer complex.  In this case things are not as pleasant  because the quetient
$$M_{k+1}/M_k $$
is a wedge  of $k$-dimensional spheres, one sphere for each critical point of index $k$.  In particular, unlike the(co)hological case, the $E_1$-sheet in the $K$-theoretical case is nontrivial in infinitely many places. Additionally, the boundary maps  in the $K$-theoretic case   do not have as nice an interpretation  as the boundary maps in the homological case. 
