# Expositions of Stallings's fibration theorem

In his famous paper

Stallings, John, On fibering certain 3-manifolds. 1962 Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) pp. 95–100 Prentice-Hall, Englewood Cliffs, N.J.

Stallings proves a theorem that (roughly stated, I'm ignoring some hypotheses) says that if $$M$$ is a 3-manifold, then every short exact sequence $$1 \longrightarrow G \longrightarrow \pi_1(M) \longrightarrow \mathbb{Z} \longrightarrow 1$$ with $$G$$ finitely generated comes from a fiber bundle $$M \rightarrow S^1$$. In particular, $$G$$ is a surface group.

This is an oft-quoted theorem, but I have trouble reading the paper. Are there any expository accounts of it anywhere?

• You may find this recent MO question about Stallings’ paper useful. mathoverflow.net/q/377412/1463
– HJRW
Oct 1, 2021 at 7:19
• Good question. I've spent time today looking, and I have failed to find an exposition. I'm sure that I've seen one (perhaps a master's thesis?) at some point... In any case, you could perhaps say where you are stuck, and we could (try to) help? Oct 1, 2021 at 21:04
• @HJRW: Thanks! That clears up some of the confusing parts, and indeed I think I now know how to prove that $G$ is a surface group carried by a surface in $M$ (the first half of the paper). If no one finds an exposition, I'll try to isolate a specific question. Oct 1, 2021 at 21:05