Euler Class constant on Fibered Face of Unit Thurston Norm Ball?

Question

I am reading about the Thurston norm out of Candel and Conlon's "Foliations 2" and Calegari's "Foliations and the Geometry of 3-Manifolds".

I'm trying to work through the proof of the "Fibered Faces" Theorem. I will denote the Thurston norm by $\xi$ throughout this post.

I understand Calegari's Lemma 5.14: if you have a "seed fibration" (that is, a fibration of $M$ over $S^1$ with surface $S$ and $\chi(S) < 0$), it follows that $\xi(\bullet) = |e(T\mathcal{F}) \cap \bullet \ |$ holds on a neighborhood of $[S]$ in $H_2(M; \mathbb{R})$. This proof essentially relies on the fact that the Euler class pairing $\langle e(T\mathcal{F}), \bullet \rangle$ spits out an integer, so it must be constant in a small $\epsilon$-ball around the seed fibration in $H_2(M; \mathbb{R})$.

However, the next theorem (Calegari's Theorem 5.15, which is the Fibered Faces Theorem), seems to imply that this pairing $\langle e(T\mathcal{F}), \bullet \rangle$ is constant on a fibered face: he states that given a fibered ray with associated foliation $\mathcal{F}$, and $S$ some Thurston norm minimizing surface on a ray intersecting that same face $F$, $$e(T\mathcal{F}) \cap [S] = \xi([S]) = -\chi(S)$$ The assumption that $S$ is in the $\epsilon$-ball doesn't arise.

Why does this equality hold? Is $\langle e(T\mathcal{F}), \bullet \rangle$ actually constant on this face?

I am trying to reconcile this statement with the one in Candel and Conlon, since they are very careful to specify that the Euler class is constant in a cone of the $\epsilon$-ball.

Any insights are sincerely appreciated!

Answer 1

Have a look into Fried's Exposé: "Fibrations over $S^1$", particularly the proof of Theorem 14.6. He proves first the linearity on a neighborhood, which you understood already, then shows the linearity on the cone over a fibered face, moreover the Thurston map is given by the Euler class $e(T\mathcal F) \in H^2(M;\mathbb R)\cong H_1(M,\partial M;\mathbb R)$, as pointed out by Ian Agol in a comment considered as a linear functional on $H^1(M)$. The proof relies on isotoping a minimal representant so that its component lie in a leaf of the foliation or only have saddle tangencies with it. The Hopf Index Theorem gives you a relation of the (carefully chosen) intersections to the surface's characteristic, while the intersections can also be related to the (geometric interpretation, i.e. "cup is dual to transverse intersection", of) the evaluation under the Euler class. You should have a look at it, it's a nice read.