TITLE: Other Homology Theories still Count Holes?
QUESTION [14 upvotes]: 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 $SH_\ast(T^\ast M)=H_\ast(\mathcal{L}M)$).

REPLY [4 votes]: 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.