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Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.

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Is the homeomorphism class of a connected open set of C determined by its fundamental group?

The answer is indeed no as David Cohen has pointed out, and more generally the answer is determined by the complements of the sets U and U'. The complete solution is effectively due to R.L. Moore (19 …
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