Let $M$ be a compact manifold.
In their paper "On the Floer Homology of Cotangent Bundles", A. ABBONDANDOLO and M. SCHWARZ define the Floer homology of $T^\*M$ by looking at 1-periodic Hamiltonian orbits on the space of smooth loops on $T^\*M$ and show that the resulting Floer homology is isomorphic to the singular homology of free loop space $LM$.


However, some people define the Floer homology of $T^\*M$ by looking at 1-periodic Hamiltonian orbits in the fixed homotopy class $h$, where 


$$ h \in [S^1, T^\*M]$$


My question is: By using the above definition, is it true that the resulting Floer homology is isomorphic to the singular homology of $L\_h M$, where $L\_h M$ is the space consists of
loops in the homotopy class $h$ ( hence it is just a subset of the free loop space $LM$)?



floer homology and viterbo's theorem
I think that the Floer complex decomposes as a direct sum of over the *conjugacy classes* of $\pi\_1(M)$, no? Since the manifold on which we are doing infinite-dimensional Morse theory is of unbased, not based, loops.


In Theorem 3.1 of [this](http://arxiv.org/abs/math/0408280) version of the paper you refer to, the last assertion is:



> 
> $\Theta$ is compatible with the splitting of the Floer and the Morse
>  complex into the subcomplexes corresponding to different conjugacy
>  classes of $\pi\_1(M)$.
> 
> 
> 


Does that answer your question?


