Space form problem
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In mathematics, the space form problem is a conjecture stating that any two compact aspherical (all higher homotopy groups are trivial) Riemannian manifolds with isomorphic fundamental groups are homeomorphic.
One might wish to conjecture that the manifolds are isometric, but rescaling the Riemannian metric on a compact aspherical Riemannian manifold preserves the fundamental group and shows this to be false. One might also wish to conjecture that the manifolds are diffeomorphic, but Milnor's exotic spheres are all homeomorphic and hence have isomorphic fundamental group, showing this to be false.
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