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Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).

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What is the "right" definition of the homology(cohomology) of an orbifold?

I do not know the current state of the art, but I can point you to the paper of Chen and Ruan which should have everything you are asking for.
Michael Joyce's user avatar