TITLE: Is Floer homology always isomorphic to the singular homology of some space?
QUESTION [9 upvotes]: After I studied Morse homology, I'm now studying the following Floer homology theories : 

1) Symplectic Floer homology ;
2) Floer homology of lagrangians ;
3) Heegard-Floer homology ; 
4) Instanton Floer homology ;
5) Seiberg-Witten-Floer homology ;

I saw that Morse homology of a compact manifold $M$ is isomorphic to singular homology of $M$. 
Similarly, I wonder if 1), 3), 4) and 5) are always isomorphic to singular homology of some corresponding space. If not, under which conditions are they ?
Moreover, does 2) link in some way or another to the others ?

REPLY [3 votes]: In the case of the Floer homology of the cotangent bundle the answer is yes. You should have a look at this:
The Viterbo transfer as a map of spectra by Thomas Kragh.