 Rokhlin's theorem shows that it has no smooth structure (as does Donaldson's theorem), and in fact, combined with the work of Andrew Casson on the Casson invariant, this shows that the formula_1 manifold is not even triangulable as a simplicial complex.The manifold can be constructed by first plumbing together disc bundles of Euler number 2 over the sphere, according to the Dynkin diagram for formula_1.This results in formula_4, a 4-manifold with boundary equal to the Poincaré homology sphere.