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I would like to understand what exactly Floer homology $HF_*(Y)$ for a simply connected compact 3-manifold $Y$ is. I understand there are many variants of Floer homology (and cohomology) but I would like to ask for a "take home universal informal definition".

Of special interest, as a physicist, is the instanton-Floer homology. But what mathematical information is encoded there? Since it is a homology theory over a space such as $\Sigma_g \times S^1$, I should be able to define it solely in terms of singular homology? Or can I somehow view instanton Floer cohomology in terms of de Rham cohomology?

Most textbooks require a lot of background, so I am happy with a vague answer to help me get intuition.

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    $\begingroup$ One comment is that the three sphere is the only simply connected compact 3-manifold (without boundary). Would you want to restate the question in terms of simple integral homology spheres? $\endgroup$
    – Neil Hoffman
    Sep 19, 2019 at 19:20
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    $\begingroup$ Generators of Floer chain complex = objects defined over Y (such as connections, or embedded circles). Differential = certain count of objects on Y x R (R = reals) which are “asymptotic to” the generators on Y (as you move towards $\pm\infty$). What you should look up, that is similar in spirit, is Morse homology of a manifold. $\endgroup$
    – Chris Gerig
    Sep 19, 2019 at 23:10
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    $\begingroup$ "I would like to understand what exactly Floer homology of a 3-manifold is." So would I, and, I imagine, every other researcher in the area. There is certainly not some obvious space it is the singular cohomology of. In a few cases (not instanton!) there is a "spectrum" that Floer homology is the homology of. But the constructions are not so enlightening as you might like. $\endgroup$
    – mme
    Sep 20, 2019 at 1:40
  • $\begingroup$ @MikeMiller I have no way to know if Floer homology is a difficult topic for people working on the field or not, and I have no idea if there is a physical interpretation. Your comment, with the sarcasm it entails has no position in stack exchange. Mocking my question, even in a not mean way, is rude. As I stated I am a physicist thus I ask here for some information not a course. Look at Chris's comment, that is what I am looking for, comments in that direction. $\endgroup$
    – Marion
    Sep 23, 2019 at 9:42
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    $\begingroup$ Marion, to my eyes, there was no sarcasm or mockery. In fact, I see it as expressing sympathy and commiseration. $\endgroup$
    – Todd Trimble ♦
    Sep 23, 2019 at 10:56

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