An $n$-dimensional submanifold $L$ of a symplectic manifold $(M^{2n}, \omega)$ is called Lagrangian if $\omega|\_L = 0$. I want to get some feeling about how many Lagrangian submanifolds are.


For each $\alpha \in H\_n(M)$, is there a Lagrangian submanifold representing $\alpha$? Maybe it's not a good idea to distinguish Lagrangians by its homology classes. Floer homology is the same if we move Lagrangians by Hamiltonian isotopy. Is there a notion of space of Lagrangian submanifolds modulo Hamiltonian isotopy?



How many Lagrangian submanifolds?
The answer to the first question is 'no'. For example, if $M = S^2\times S^2$ is given the product symplectic structure, then there is no Lagrangian submanifold in the homology class of $S^2\times\{x\}$.


The answer to your second question is 'yes, the notion exists', but the question is whether this set can be endowed with any good properties that makes it into a 'useful space'. For example, just look at the closed curves on a surface $S$ endowed with a nonvanishing $2$-form $\omega$. This is the space of (closed) Lagrangian submanifolds in this case. What can you say about the set that you get by regarding two that differ by Hamiltonian isotopy as equivalent? 


