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This may be a naive question, but since first learning homology I considered it as a tool which counts appropriate holes in your space (on top of orientation and torsion phenomena). Then I was introduced to homology of groups and now Floer/Morse homologies. Do these homologies still count "holes" in some fashion?
In the case of group homology, $H_\ast(G)\cong H_\ast(BG)$, so we can view this homology as a count of holes in the Milnor construction (CW-complex assembled from points in the discrete group with the group structure).
In Floer homology we're counting holomorphic curves (flow-lines in Morse homology), but it isn't viewed as having these curves "wrap around holes", so I am not sure if this hole-detecting view of homology breaks down.

[[Edit]]: I will narrow down my question. Are there instances where I can treat $HF_\ast$ as $H_\ast$ of a particular space? For instance, I just realized that with nice conditions we have $HF^\ast(L,L)=H^\ast(L)$ in Lagrangian-Floer homology, so here it counts the holes of the Lagrangian submanifold.
Thanks to Steven Landsburg's response, we can usually find such a space (but ideally would be looking for something explicit, such as Floer homotopy type with $SH_\ast(T^\ast M)=H_\ast(\mathcal{L}M)$).

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Technically, singular homology does not quite count holes. $H_0 X$ is free abelian on the path-components of $X$, so there's one more copy of $\mathbb Z$ than the number of $0$-dimensional holes. Said another way ,if you treat a contractible space as "having no holes", then $H_0$ can't be measuring holes as it's not trivial. There's a calibration issue -- you need to take the associated reduced homology. That way, the homology theory is trivial on a contractible space. So sure, it measures holes, in that you can describe non-trivial homology classes as extension problems that... – Ryan Budney Mar 15 '12 at 4:22
have no solution. – Ryan Budney Mar 15 '12 at 4:23
This is a pretty vague question. – Andy Putman Mar 15 '12 at 5:03
@Andy, is this better? – Chris Gerig Mar 15 '12 at 7:21
up vote 21 down vote accepted

If your homology theory is of the form $H_n(X) = H_n(S(X)) $ where $S$ is some functor from your original category to non-negative chain complexes, then the Dold-Kan correspondence gives you a corresponding simplicial abelian group $\Gamma(S(X)) $ and hence (by realization) a topological space in which you are "counting holes".

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If I understand what you're proposing correctly, you give a space $Y$ where $\pi_*(Y) = HF_*(X)$ (for instance). This is much easier than the question as asked, and gives less information. See for a paper by Everitt, Lipshitz, Sarkar, and Turner comparing two such constructions for an example. – Dylan Thurston Apr 3 '12 at 9:54

Symplectic homology of the cotangent bundle is the homology of loop space (see Viterbo's "Functors and computations in Floer homology" or Abbondandolo-Schwartz).

Also, Cohen-Jones-Segal have a paper in the Floer memorial volume which outlines the construction (modulo analytical details of e.g. defining smooth structures on compactified moduli spaces) of a spectrum whose homology recovers a given Floer homology. See early work of Manolescu on the analogous problem in Seiberg-Witten-Floer homology or this paper of Lipyanskiy which extends the Viterbo-Abbondandolo-Schwartz result to the level of Floer bordism.

EDIT: I would like to say a little more about this. Cohen-Jones-Segal prove that one can construct a manifold up to homeomorphism from Morse data alone (this shouldn't be too surprising when you remember that any compact manifold admitting a Morse function with two critical points has to be homeomorphic to a sphere). So although it's true (as Steven Landsberg says) that you can construct a space by geometric realisation of a Dold-Kan construction applied to the chain complex used to define homology, it's not clear to me that this will reconstruct the original space you started with (maybe it works up to homotopy?).

The idea of Floer homotopy theory is therefore strictly deeper than just 'constructing a space whose homology gives you Floer homology'. It should really give new Floer theoretic invariants (e.g. the work of Barraud and Cornea on the 'quantum' Serre spectral sequence).

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regarding the lack of analytical details in the CJS paper, you might want to check out the work of Lizhen Qin. He is at Purdue now, and his thesis was precisely on this issue of smooth structures and compactifications. – Sean Tilson Mar 15 '12 at 21:09
Ah yes, Michael Hutchings mentioned this to me along with references but I have not checked it out yet. – Chris Gerig Mar 16 '12 at 2:10

Similar to the Floer homotopy type results Jonny cites, there are a few recent papers by Lipshitz-Sarkar on constructing a spectrum whose (singular) homology is Khovanov homology.

Besides giving an alternate construction of these various homologies, and the intrinsically interesting question of what spaces/spectra might underlie (or at least be related to) the Floer-theoretic constructions, I think there was some hope that other topological invariants of the spectra (e.g. generalized homology theories) would give new interesting invariants attached to, say, the underlying 3-manifold (in the Seiberg-Witten case). But I don't know whether anything along these lines ever panned out.

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In finite dimensional Floer homology, the connecting trajectory count encodes a more traditional count. Namely the unstable manifolds of the flow define, under certain assumptions, a cellular decomposition of the manifold. The associated cellular chain complex is isomorphic to the Floer complex. The isomorphism associates to each critical point, viewed as an element in the Floer complex, its unstable manifold, regarded as an element of the cellular complex. This interpretation is meaningless in infinite dimensions.

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