# How decomposition of prime 3-manifold along incompressible tori help in classification of 3-manifolds

We know about geometrization conjecture. I am trying to understand how decomposition of prime manifold along compressible tori help classify 3-manifolds. Can we construct list of all prime manifolds this way ?

Once we decomposed given prime manifold $M$ on components having tori boundary we should been able to glue up the components back to obtain $M$. Therefore I thought that it is possible to list all possible components $C_1, C_2,...$ each of them having tori boundary (and no more incompressible torus) and have recipe of gluing such that any prime manifold can be obtained this way. This procedure should be so good, so for each prime $M$ there should be different pair (components, gluing recipe). Obviously we have already such procedure - it is surgery on link embedded in $S^3$, but is this 3-manifolds classification ? Maybe my knowledge is little here. Do we know when prime manifold is obtained as result of surgery and when two links give the same prime manifold ? If yes, then it remains to list all possible links in $S^3$.

Related question is what can be the genus of Heegaard splitting of prime 3-manifold ?

• it's not some random decomposition along incompressible tori like you would get from a surgery presentation, but the minimal one (JSJ decomposition) or the geometric decomposition so that each piece is Seifert-fibered or hyperbolic (so that the pieces are well-understood). Perhaps the most important point is that the group isomorphism problem is solvable for hyperbolic groups. – user98602 Jul 2 '18 at 8:41
• I see and what it is the recipe for building the list of prime 3-manifolds from well understood pieces ? – Marek Mitros Jul 3 '18 at 9:23
• arxiv.org/pdf/1508.06720.pdf – user98602 Jul 4 '18 at 2:58
• Thank you, I am reading. Although my doubts still remains. I understand the decomposition of arbitrary closed manifold onto connect sum of prime manifolds. Next I do not follow the purpose of JSJ decomposition along incompressible tori. But assuming we have it. What is the procedure in obtaining any prime closed 3-manifolds ? Do we know all possible pieces with tori boundary which we can glue to prime manifold ? How can we glue them ? Is resulting manifold dependant on the "gluing method" i.e. on the torus homomorphism we choose ? I do not find answers on all my questions, but I am reading... – Marek Mitros Jul 4 '18 at 14:10
• I meant "torus homeomorphism" above... – Marek Mitros Jul 4 '18 at 14:34