The minimal resolution S also contains a copy of the Poincaré homology sphere, namely π^-1(SU(2)/Γ). The reason is that π is an isomorphism away from the singularity.

But this copy of the Poincaré sphere bounds a smooth manifold with boundary, M, since S is smooth!

M ⊂ S

∂M = π^-1(SU(2)/Γ)

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