I am looking for a reference for the **Stallings-Epstein-Waldhausen construction** (constructing an incompressible surface in a 3-manifold from a nontrivial splitting of the fundamental group).


I know of Proposition 2.3.1 in the following article, but was hoping for a more comprehensive / textbook-like treatment.


Marc Culler and Peter B. Shalen: Varieties of Group Representations and Splittings of 3-Manifolds. Annals of Mathematics, Vol. 117, No. 1 (Jan., 1983), pp. 109-146.


My goal is to find out under which conditions the resulting incompressible surface is orientable.



Reference request: Stallings-Epstein-Waldhausen construction
As an answer to your first question: You can find an exposition of the pull-back construction in Scott's notes "[An introduction to 3-manifolds](https://www.math.cuhk.edu.hk/course_builder/1617/math6071b/scott_An%20Introduction%20to%203-manifolds-1974.pdf)".


As an answer to your second question: If the group we are splitting over is not free, then we can detect the orientability of the surface by abelianising the group. If the group is free, then we are obtaining a surface with boundary. As HJRW notes, the surface obtained is always two-sided. Thus it suffices to know that $M^3$, the ambient three-manifold, is orientable. If $M^3$ is not orientable, there is still a way to detect the orientation of the splitting surface, by lifting to the orientation double cover and seeing how much of the edge group lifts.


