I started reading the book *"Lagrangian intersection Floer theory anomaly and obstruction"*, and there are 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 interested 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.Just define $A(\gamma)=\int_{0}^{1}\gamma^* \lambda_{can}$.

But then how is the index of a critical point $p$ defined ? We could use the more general definition that they talk about but then we would to pick a path $l_0$ and an homotopy class $[w]$ ,so I am wondering if this will be equal to something simpler and more intuitive ?

My motivation for this simpler case comes from the following: Let's consider our symplectic manifold to be the cotangent bundle and fix the fibers $T^*_{q_0} M, T^*_{q_1}M$. Now suppose we have an Hamiltonian flow $\phi^t$, from an hamiltonian function $H$, such that $\phi^1(T^*_{q_o}M)\pitchfork T^*_{q_1}M$. In the following paper https://arxiv.org/pdf/math/0408280.pdf there is developed an isomorphism between the Floer Homology of the action functional whose critical points are orbits of our hamiltonian vector field and the corresponding Lagrangian functional on $TM$. Then one also defines here the index $\mu_{\Omega}(x)$ where $x$ is an orbit of $X_H$,and is able to prove its the same has the morse index of the action functional of the Lagrangian. And so for example in the case where $H(q,p)=\frac{1}{2}|p|^2$, and we get the Lagrangian $L(q,v)=\frac{1}{2}|v|^2$ we will get that $\mu_{\Omega}(x)$ will give us the index between $x(0)$ and $x(1)$. And so I am wondering if we look at the lagrangian intersection theory of $\phi^1(T^*_{q_0}M)$ and $T^*_{q_1}M$ and we looked at the intersection point that is $x(1)$, we will get that it's index is the same as $\mu_{\Omega}(x)$? If this were true it would be a nice geometrical interpretation of the index relating to geodesics and Jacobi fields.

Any insight is appreciated. Thanks in advance.