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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?

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    $\begingroup$ Associated to any such $M$ and $f$ there is a topological category, $C_f$, whose realization is homeomorphic to $M$ (the objects are critical points and the morphisms are flows between them, basically). I imagine that continuous functors to the category with one object and $U$ as its morphism space might give $K$-theory? Unwinding we could probably give a more geometric description... Anyway, this is all sort of a guess. $\endgroup$
    – Dylan Wilson
    Commented Apr 3, 2013 at 14:59
  • $\begingroup$ (Oh I guess I should say it's only a homeomorphism if $f$ is generic enough... in general it's a homotopy equivalence.) $\endgroup$
    – Dylan Wilson
    Commented Apr 3, 2013 at 15:02
  • $\begingroup$ I always wanted to know the answer to this question, ever since reading Cohen-Jones-Segal: "Floer's infinite dimensional Morse theory and homotopy theory" where they construct the flow category Dylan Wilson mentioned. You could ask the same question about your favourite generalised cohomology theory. $\endgroup$
    – Jonny Evans
    Commented Apr 3, 2013 at 17:15
  • $\begingroup$ What is your "critical point data"? Just the index isn't enough, right? I'm thinking about the blowup of CP^2 vs. S^2 x S^2, two nonhomotopic manifolds on which I have Morse functions with the same indices 0,2,2,4. Or were you ruling out the repetition of 2? $\endgroup$
    – Allen Knutson
    Commented Apr 3, 2013 at 17:47
  • $\begingroup$ "Flow data" would be a better description. You need to know stuff like the homotopy type of the attaching map of the cell. Even for Morse homology you need flow data for the differential. I seem to recall that Cohen-Jones-Segal explain what you would need to reconstruct the stable homotopy theory in terms of manifolds of flow-lines. They pick this case because it's all you can hope to recover in Floer theory - having infinite-dimensional stable/unstable manifolds means that the "homotopy type of the attaching map" wouldn't be very interesting even if it made sense. $\endgroup$
    – Jonny Evans
    Commented Apr 3, 2013 at 19:43

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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 ([email protected]).

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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.

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