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 $SH_\ast(T^\ast M)=H_\ast(\mathcal{L}M)$).

reducedhomology. 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