On a special manifold $\mathbb{C}$ the well known Monodromy theorem holds. My question is if it holds for any complex 2-dimensional manifolds as well?
More concretely:
I can show: the first (more trivial) part namely that the analytic continuation does not depend on chosen cover of a given curve.
It is left to show: that if we assume that analytic continuation exists along any curve in $M$ and $H: I \times I \rightarrow M$ is a homotopy between $H(0,\cdot)$ and $H(1,\cdot)$ than value obtained at $H(0,1)=H(1,1)$ is the same.
I would appreciate a precise approach (since I have actually a problem in detailing the proof: my approach is to show that we have a well defined function on $H(I \times I)$ by considering the times for which we have a well defined function on $H(I\times [0,s])$ ) or reference.