1
$\begingroup$

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 $LM$)?

$\endgroup$

1 Answer 1

2
$\begingroup$

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?

$\endgroup$
1
  • $\begingroup$ I see, hence there is indeed a version of Viterbo's theorem when defining the Floer homology of $T^*M$ by looking at the free loops in the conjugacy class of $\pi_1(T^*M)$. Thanks! $\endgroup$ Dec 12, 2014 at 2:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.