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In section 3 of his paper "Mappings of reducible 3 manifolds" McCullough, proves that every self-homeomorphism of a reducible 3 manifold can up to isotopy be written as a composition of particularly nice homeomorphisms, namely permutations of homeomorphic factors, automorphisms of factors, flips of $S^2 \times S^1$ factors and so called slide homeomorphisms.

Does there exist a similar result for homeomorphisms of 3-manifolds with boundary i.e. can a self-homeomorphism of a 3-manifold with boundary be decomposed into similarly nice parts?

As Sam Nead pointed out in this generality this is too much to ask. Does something similar hold if one requires the homeomorphism to be the identity on the boundary?

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    $\begingroup$ The McCullough result doesn't really describe the automorphisms of reducible 3-manifolds -- if you don't know the automorphism group of the reducible factors. i.e. it reduces to another case, which you have to deal with using other tools. At that level of generality, yes there's plenty of results. If the manifold has boundary there is the extension $Diff(M) \to Diff(Filled M) \to Emb(Filling manifold, filled M)$ where the "filled M" means any closed manifold you can construct by attaching handlebodies to the boundary of $M$, and the space on the right is the space of embeddings. . . $\endgroup$
    – Ryan Budney
    Commented Jun 3, 2023 at 0:43
  • $\begingroup$ of the handlebody in the filled manifold. That would be a type of generalization of McCullough's result. $\endgroup$
    – Ryan Budney
    Commented Jun 3, 2023 at 0:44
  • $\begingroup$ @RyanBudney maybe my wording was a bit off. I'm aware that it does not describe the automorphisms completely but it splits it into more manageable pieces that are mostly related to automorphism groups of prime manifolds. For the application I had in mind I wanted something similar since I can manage automorphisms of prime manifolds just fine but the reducible case seems too complicated without a McCullough type result. $\endgroup$
    – ThorbenK
    Commented Jun 3, 2023 at 4:52

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Some care is required in phrasing the question (and the desired form of the answer). For consider the following two examples.

  1. Suppose that $M = V_g$ is the handlebody of genus $g$. For example, $M$ may be obtained as a small closed regular neighbourhood of a connected finite graph nicely embedded in three-space. The (orientation preserving) mapping class group of $M$ is called the genus $g$ handlebody group. It is generated by handle slides and Dehn twists about disks.

  2. Suppose that $M = S_g \times I$ is the product of the surface of genus $g$ and an interval. Then the (orientation preserving) mapping class group of $M$ is (up to finite index) a copy of the mapping class group of $S_g$. The latter is generated by Dehn twists about curves, so the former is generated by Dehn twists about annuli.

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  • $\begingroup$ Those are nice examples! I further specified the question. Does something similar work if one requires the homeomorphism to be the identity on the boundary? $\endgroup$
    – ThorbenK
    Commented Jun 2, 2023 at 14:41

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