These questions are ok but it is important to understand as much as you can each of the categories:

• homotopy types satisfying poincaré duality
• topological manifolds
• sobolev manifolds eg quasiconformal or bilipschitz
• piecewise linear or piecewise differentiable C1, C2, ... Cr.....Cinfinity
• 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

These questions are ok, but it is important to understand as much as you can about manifolds. For each of the categories: a] Homotopy Types satisfying Poincaré Duality... b] Topological Manifolds... c] Sobolev Manifolds eg Quasiconformal or Lipschitz Manifolds... d] Piecewise Linear or Piecewise Differentiable Manifolds... e] C1, C2,...C-infiniy = Smooth Manifolds... f] Real Analytic Manifolds... g] Real Algebraic Manifolds... The types with canonical coordinates::: h] Poisson Manifolds... i] Symplectic Manifolds... j] Complex Manifolds... k] Generalized Complex-Symplectic manifolds...
l] Geometrized Three-Manifolds... One knows contexts where each of these categories are particularly useful. [Dennis Sullivan]

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

These questions are ok but it is important to understand as much as you can each of the categories:

• homotopy types satisfying poincaré duality
• topological manifolds
• sobolev manifolds eg quasiconformal or bilipschitz
• piecewise linear or piecewise differentiable C1, C2, ... Cr.....Cinfinity
• 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