Fake projective plane
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For Freedman's example of a non-smoothable manifold with the same homotopy type as the complex projective plane, see 4-manifold.
In mathematics, a fake projective plane (or Mumford surface) is one of the 50 complex algebraic surfaces that have the same Betti numbers as the projective plane, but are not isomorphic to it. Such objects are always algebraic surfaces of general type.