Q:

Is there a biholomorphic map between a simply connected domain and non simply connected domain?

Is there a biholomorphic map between a simply connected domain and non simply connected domain?

I am not sure how to approach this question. This is not a homework question but one I simply came across. Perhaps we would be wise looking at the number of homology basis? Does Riemann mapping theorem work here? I have little idea of how to approach this question. I think that the open mapping theorem would be a good use, but I am not sure how to use it.

A:

Simple connectedness is a topological invariant, and biholomorphisms are homeomorphisms. Therefore the answer is no.

