On JSJ theory for Haken 3-manifolds
by Tharatorn Supasiti
Abstract: The Torus Theorem states that if a compact, orientable, irreducible 3-manifold admits a singular essential torus, then it either admits an embedded one or it is a small Siefert fibred space. It is one in a series of beautiful results that explore the interplay between algebra and topology. The first published proof was given by Feustel in 1976. It relies heavily on the JSJ theory that was developed by Jaco and Shalen, and independently by Johannson. So, in a way, both theories should be viewed together. Here, the existence of a singular essential torus allows us to explicitly construct the characteristics submanifold $K$ from which the desired embedded essential torus is contained in its boundary or $K$ is the whole manifold itself. The question here is whether or not we can develop one for Haken 4-manifolds as defined by Foozwell. As an exercise, we gave a short proof that uses Aitchison and Rubinstein's construction.
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