Floer boundary-operator conjecture for connecting manifolds

Let $C_k(R)$ be the free $R$-module generated by the $k$-dimensional connecting manifolds of

. **Floer boundary-operator conjecture.** For a generic \subset of coefficients $a_i$ in

, there are maps

$$\partial_k:C_k\to C_{k-1}$$

such that each $\partial_k$ is an $R$-module homomorphism, $\partial_0=0$, $\partial_{k-1}\circ\partial_k=0$, and $v=\partial_k(u)$ precisely when, in the stated geometric sense, $v$ is a sum of boundary elements of $u$ obtained by a deformation retraction of a neighborhood of $v$ in $u$ onto $v$. This would produce the Floer complex; the source describes the construction as an involved, unresolved gluing theorem.