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Can anyone explain something in Gabai's foliations and topology of 3-manifolds III. On page 492 where he proved the existence of an essential surface, I do not understand his diagram for the compressing disk.

Hesky Cee
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I'm assuming you mean figure 4.2(b). From a brief glance without knowing the rest of the context, here's my interpretation: you're trying to prove that $P\cap Q$ only contains essential arcs after isotopy, and this step deals only with inessential arcs $\lambda\in P\cap Q$ which bound disks $D\subset P$ which contain no other inessential arcs. 4.2(b) is just a "side view" of such a disk, which is given a "top view" in 4.2(a).

gmoss
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  • Ofcourse those are the side and top views. Did you notice the small circle in the side view? The top view shows that it is the intersection of D with Q (I hope I'm right). If that's the case, why can't we remove that intersection by isotopy? When this is sorted, I have a more serious question to ask :) – Hesky Cee Oct 03 '13 at 21:32