Is there any paper where I can find a good explanation of the JSJ decomposition, the geometrization theorem and the relations between them when the manifold has nonempty (and non necessarily toroidal) boundary? Currently I am trying to read the original Jaco-Shalen paper and a Scott's survey paper, buy they do not look to be completely satisfactory...
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8$\begingroup$ See Bonahon's survey www-bcf.usc.edu/~fbonahon/Research/Preprints/Handbook.pdf in [Handbook of Geometric Topology (R. Daverman, R. Sher eds.), Elsevier, 2002, pp. 93-164]. $\endgroup$– Igor BelegradekCommented Aug 17, 2013 at 20:06
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$\begingroup$ If you have boundary there is an extra layer in the geometric decomposition of manifolds that Bonahon describes in the paper Igor cites, called the compression-body decomposition. $\endgroup$– Ryan BudneyCommented Aug 17, 2013 at 22:33
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4$\begingroup$ Neumann has some notes on 3-manifold topology which includes a section on the JSJ decomposition.ams.org/mathscinet-getitem?mr=1747270 $\endgroup$– Ian AgolCommented Aug 18, 2013 at 3:09
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The work of Jaco-Shalen and Johannson actually handles manifolds with boundary. The theorem they prove (in the language of Johannson's Book) is:
An irreducible, boundary-irreducible 3-manifold with useful boundary pattern possesses a characteristic submanifold, unique up to isotopy.
The complement of the characteristic submanifold is atoroidal and acylindrical, hence (by Thurston) possesses a unique hyperbolic metric with totallygeodesic boundary.
The characteristic submanifold has the property that every admissibly embedded I-bundle or Seifert fibered space is isotopic into the characteristic submanifold.
All the notions, including useful boundary pattern and admissible embedding, are explained in Springer Lecture Notes in Mathematics 761 (Johannson: Homotopy equivalences of 3-manifolds with Boundaries), the above-cited Theorem is Proposition 9.4 in that Book.
