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My apologizes if this is a fairly elementary question, I am still a novice when it comes to 3-manifold topology.

I am wondering the following: by Kirby calculus, we know that two links (say in $S^{3}$ related to each other via Kirby moves give rise to homeomorphic 3-manifolds -- and the converse is true as well: homeomorphic 3-manifolds can be given surgery presentations related to each other via Kirby moves.

If $M$ and $N$ are homotopic 3-manifolds, what do we know about the relationship between their possible surgery presentations?

Thank you!

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  • $\begingroup$ For hyperbolic 3-manifolds, at least, Mostow rigidity gives that homotopy-equivalence implies homeomorphism. $\endgroup$
    – Neal
    Oct 21 at 15:26

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