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In dimension 3, we have the celebrated Kirby theorem: Let $L_1, L_2$ be two links in the 3-sphere $S^3$; then they surgeries along them produce homeomorphic 3-manifolds if and only if they are related by blow-ups and blow-downs.

I wonder if such wonderful theorem has a 4-dimensional generalization. We do have basic moves for the Kirby diagrams from handle slidings and cancellations, but do they form a complete set of moves that characterize smooth 4-manifolds?

If such question is too hard, what can we say about for the more restricted cases (e.g. connected smooth closed oriented 4-manifold, whose Kirby diagram can be made simpler)?

Related

  1. What are Kirby diagrams of candidate exotic 4-manifolds?
  2. Given a Kirby diagram of a 4-manifold, what's its homotopy 2-type?
  3. Can cobordisms of 3 or 4 manifolds be visualized by moves on kirby diagrams?
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    $\begingroup$ Yes, Cerf's theorem holds in all dimensions (even relative to a boundary) and is stated as 4.2.12 in the book by Gompf and Stipsicz. I think it is time to read that book :) $\endgroup$
    – mme
    Sep 24, 2021 at 16:07
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    $\begingroup$ However, Cerf's result is not a generalization of Kirby's theorem. $\endgroup$
    – user101010
    Sep 24, 2021 at 16:10
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    $\begingroup$ It may help to bear in mind that there is no algorithm to tell if two smooth closed 4-manifolds are diffeomorphic. Also, Kirby's theorem is a uniqueness result for a completely different sort of manifold description. Namely, it describes when two Dehn surgery descriptions are the same - these are not the same as handlebody decompositions. $\endgroup$
    – user101010
    Sep 24, 2021 at 17:03
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    $\begingroup$ @Student: Just because you have a set of moves that connect two objects if they happen to be isomorphic does not mean you have an algorithm to decide if those objects are isomorphic. For this, you would need an a priori bound on how many moves are needed. So e.g. Kirby's theorem about 3-manifolds does not give you an algorithm to tell if two 3-manifolds are homeomorphic. It gives you an algorithm that will eventually terminate of they are homeomorphic (ie enumerate all possible Kirby moves), but will run forever if they are not. $\endgroup$
    – Linda
    Sep 24, 2021 at 18:20
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    $\begingroup$ (it is true that there is such an algorithm for 3-manifolds, but it is far more complicated. in other contexts, there might not be an algorithm at all. e.g. it is known that if two group presentations define the same group, then they can be connected by a sequence of Tietze transformations [in fact, this is almost trivial]; however, it is also known that there is no algorithm to decide if two group presentations define the same group.) $\endgroup$
    – Linda
    Sep 24, 2021 at 18:23

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