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these questions are ok but it is important to understand as much as you can Each of the categories homotopy types satisfying poincaré duslity topological manifolds sobolev manifolds eg quasiconformsl or bilipscjitz piecewise linear or piecewise differentiate C1, C2, ... Cr.....Cinfiniy real analytic real algebraic then the multiple types with canonical coordinates complex manifolds symplectic manifolds in dimension three those with coordinates so that overlap maps satisfy a global analytic continuation property one knows contexts where each of these are particularly useful. dennis sullivan