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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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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. –  Dylan Wilson Apr 3 '13 at 14:59
(Oh I guess I should say it's only a homeomorphism if $f$ is generic enough... in general it's a homotopy equivalence.) –  Dylan Wilson Apr 3 '13 at 15:02
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. –  Jonny Evans Apr 3 '13 at 17:15
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? –  Allen Knutson Apr 3 '13 at 17:47
"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. –  Jonny Evans Apr 3 '13 at 19:43

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