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For d=3 the homotopy groups can be pretty elaborate. Consider the connect-sum of some lens spaces. The universal cover embeds in S^3 as the complement of a cantor set (except for a few degenerate cases where you have RP^3 summands). So the homotopy-groups are pretty complicated (\pi_2 is finitely generated over \pi_1). You could probably make an argument that this is about the worst thing that can happen for the homotopy-groups of 3-manifolds.