In Lagrangian Floer theory, we can define an A-infinity algebra. It is by first choosing a subset $X_L$ of chains in the Lagrangian submanifold $L$, and then defining boundary maps on(Actually, sum of tensors of) chains in it . A part of its definition is, for a chain $(P,f)$, $\beta \in \pi^2(M,L)$, $$ \left\{ \begin{aligned} m_{1,\beta}(P,f)=\mathcal{M}_{2}^{main}(\beta,P)^s,& \text{ when } \beta\neq \beta_0 ,\\ m_{1,\beta}(P,f)=(-1)^P\partial{P}& \text{ when }, \beta=\beta_0 \end{aligned} \right\} $$

Clearly, it can be view as an quantum version of classical homology theory.

On the other hand, singular homology theory have intersection theory and Poincare duality, say, we can define intersection numbers by elements from $H_k(A)$ and $H_{n-k}(A)$ of a smooth manifold. This kind of theory if helpful when studying submanifolds.

Do A-infinity algebra in Floer theory has such kind of intersection theory? Furthermore, it is possible to first define a A-infinity "coalgebra" on cochains, and get a kind of Poincare duality?

A more precise and special question is, for $a,b\in HF(L)$, can we define the intersection number of $a$ and $b$?

In Lagrangian Floer theory, we can define an A-infinity algebra. It is by first choosing a subset $X_L$ of chains in the Lagrangian submanifold $L$, and then defining boundary maps on(Actually, sum of tensors of) chains in it . A part of its definition is, for a chain $(P,f)$, $\beta \in \pi^2(M,L)$, $$ \left\{ \begin{aligned} m_{1,\beta}(P,f)=\mathcal{M}_{2}^{main}(\beta,P)^s,& \text{ when } \beta\neq \beta_0 ,\\ m_{1,\beta}(P,f)=(-1)^P\partial{P}& \text{ when }, \beta=\beta_0 \end{aligned} \right\} $$

Clearly, it can be view as an quantum version of classical homology theory.

On the other hand, singular homology theory have intersection theory and Poincare duality, say, we can define intersection numbers by elements from $H_k(A)$ and $H_{n-k}(A)$ of a smooth manifold. This kind of theory if helpful when studying submanifolds.

Do A-infinity algebra in Floer theory has such kind of intersection theory? Furthermore, it is possible to first define a A-infinity "coalgebra" on cochains, and get a kind of Poincare duality?

A more precise and special question is, for $[a],[b]\in HF(L)$, can we define the intersection number of $[a]$ and $[b]$?

In standary homology theory, we first take representation $a,b$ and take fibre product of $a$ and $b$, finally say it it invariant for choice of $a,b$. It is possible to do so in $A_{\infty}$ algebra sense?