Let $A$ be open, connected set in $\mathbb{C}$. We know that if a $\alpha$ loop is null-homotopic in $A$ then it is null-homolog. Is the converse true?
Null-homolog loop in region
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complex-analysis
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0I don't know the answer for open, connected subsets of $\mathbb{C}$, but in general, the answer is no. See, for example, the Poincare dodecahedral space. – 2012-11-06