floer homology and viterbo's theorem

Question

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 $LM$)?

Answer 1

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 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?