We describe a new approach to the canonical decompositions of 3–manifolds along tori and annuli due to Jaco–Shalen and Johannson (with ideas from Waldhausen) – the so-called JSJ–decomposition theorem. This approach gives an accessible proof of the decomposition theorem; in particular it does not use the annulus–torus theorems, and the theory of Seifert fibrations does not need to be developed in advance.
Walter D Neumann. Gadde A Swarup. "Canonical decompositions of 3–manifolds." Geom. Topol. 1 (1) 21 - 40, 1997. https://doi.org/10.2140/gt.1997.1.21