Criterion to establish if there is a biholomorphic function between 2 sets in $\mathbb{C}$ The Riemann mapping theorem states in particular that if $\emptyset \neq \Omega \neq \mathbb{C}$ and $\Omega$ is open and simply connected, then there is a biholomorphic function $f:\Omega \longrightarrow \{ z \in \mathbb{C}: |z|<1 \} $. It is then easy to see that there is a biholomorphism between any two sets $\Omega_1$, $\Omega_2$ satisfying the assumptions.
Is it true that if $\Omega_1$ satisfies all assumptions, and $\Omega_2$ satisfies all the assumptions except for the simply connectedness, then there can't exist a biholomorphism between $\Omega_1$ and $\Omega_2$? I can't prove or disprove this.
I noted that for example if we pick a point $a$ in a "hole" in $\Omega_2$ then every function on $\Omega_1$ has a primitive and this does not hold for $\Omega_2$ anymore (for example $\Omega_2= B_1(0) - \{0 \}$ and $f(z)=\frac{1}{z}$ which I think can be generalised to any not simply connected set ).
Im studying complex analysis now and I don't have a good topological background.
Are there good general criteria to determine if there exist a biholomorphism between $2$ given sets?
 A: The fundamental group provides some necessary conditions. As you say in the OP, if I take one region with a hole in it and one without, they have different complex analytic properties. This will clearly prevent a bihilomorphism. One simple reason is because a biholomorphism is necessarily a homeomorphism. In fact, even a diffeomorphism, because holomorphic functions are not just complex differentiable but analytic. Another reason is that biholomorphic maps should preserve the existence of primitives, basically by the chain rule.
Also as you note, the Riemann mapping theorem says that in the case where $\Omega_1$ and $\Omega_2$ are simply connected, there are no further complex analytic invariants - they are all biholomorphic to one another. On the other hand, the story is more complicated for non-simply connected domains.
There is a version of the Riemann mapping theorem for regions with holes in them. These are sometimes called 'multiply connected' regions I think. It is a topological theorem that the only possible fundamental groups of connected planar regions with regular boundaries are free products of $\mathbb{Z}$, because the only such regions are homotopy equivalent to wedges of circles. Ahlfors proves in his Complex Analysis book the following theorem:
Given a multiply connected region $\Omega \subset \mathbb{C}$, there is a biholomorphism $F$ mapping $\Omega$ to a multiple slit region, which means an annulus with $n-2$ angular arcs removed each at some radial distance from the center and of some angular length.
The radii of the overall annulus, and the length of the slits are determined by studying the Dirichlet problem on $\Omega$, but can in some cases be determined explicitly. There is a lot we don't know about finding explicit biholomorphisms between regions, so it can be difficult to tell in general if two regions are biholomorphically equivalent. But the upgraded version of the theorem would state that two multiply connected regions are biholomorphically equivalent if they have the same invariants, in the sense that they both map to the same multiple slit region. There is a lot we don't know about finding explicit maps between regions, so I think I will stop here.
A: If $\Omega_1$ is simply connected and if there is a biholomorphic mapping $f$ from $\Omega_1$ onto $\Omega_2$, then $f$ is a hemeomorphism, and therefore $\Omega_1$ and $\Omega_2$ have the same topological properties. So, since being simply connected is a topological property, and since $\Omega_1$ is simply connected, $\Omega_2$ is simply connected too.
