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For questions about Heegaard-Floer homology (as introduced by Ozsváth-Szabó in 2003) and its uses in 3- and 4-dimensional topology.

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Questions on poincare homology spheres and branched covers

As Ian Agol points out the Orbifold Theorem will answer question 1 affirmatively. Although the original result is due to Thurston, the common references in the literature are: Boileau, Michel, Sylva …
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