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It is known that every orientable 3-manfiold can be obtained as a ramified cover of S³ with a ramification (of some order) at a link in S³. I am curious if there is a reasonable characterization of 3-manifolds that cover 3-torus?

It is known that every orientable 3-manfiold can be obtained as a ramified cover of S³ with a ramification (of some order) at a link in S³. I am curious if there is a reasonable characterization of 3-manifolds that cover 3-torus?

Added. Notice that such a manifold is enlargeble, so it does not admit a metric of positive scalar curvature, so for example a connected sum of n copies of S² x S¹ does not admit a ramified cover of T³ (as far as I understand).

It is known that every orientable 3-manfiold can be obtained as a ramified cover of S^3 with a ramification (of some order) at a link on S^3. I am curious if there is a reasonable caracterisation of 3-manifolds that cover 3-torus?

It is known that every orientable 3-manfiold can be obtained as a ramified cover of S³ with a ramification (of some order) at a link in S³. I am curious if there is a reasonable characterization of 3-manifolds that cover 3-torus?