I started reading the book "Lagrangian Intersection Floer Theory Anomaly and Obstruction", and there a couple of details and assumptions in the definition of the Novikov covering that I would like to clarify. (I will follow the notation that they use in the book )

First is why do we need the actual use of the Novikov covering ? With this I mean why are we asking for two conditions on that is for the integrals of the symplectic areas to be the same , I guess this makes sense so that the action functional is well-defined, but wouldn't this happen in the universal cover since the symplectic form is closed? I guess I don't understand why we need/want the extra assumption $I_{\mu}(\bar w \# w´)=0$. Will it be for the Maslov index of a point $[l_p,w]$ to be well-defined ? (At least it's the only thing I could see that it's useful for).

Then I am interest in symplying this and consider just exact symplectic manifolds $(M,d\lambda)$. Now here we are able to define the action functional in $\Omega(L_0,L_1)$ and still get that the intersection points are the critical points of this action functional. But then how is the index of a critical point $p$ defined ? We could use the more general definition that they talk about , but I am wondering if this will be equal to something simpler and more intuitive ?

Any insight is appreciated. Thanks in advance.