Floer homology conjecture for heteroclinic-orbit spaces

Let $E$ be the space of heteroclinic orbits of

, and let $F_k$ denote the homology modules of the proposed Floer complex, $F_k=\operatorname{ker}\partial_k/\operatorname{im}\partial_{k+1}$. **Floer homology conjecture.** For a generic \subset of the coefficients $a_i$ in

,

$$F_k\cong H_k(E;R).$$

Thus the Floer complex should compute the homology of the space of heteroclinic orbits. The source presents this as a reasonable conjectural outcome of the still-unconstructed gluing theory.