Thurston Norm

Abstract

In this chapter, we prove the following theorem. Theorem 2.1 (W. Thurston [Thu85]). Suppose that M is a compact atoroidal orientable 3-manifold such that rank $H_2 (M, \partial M; Z) \leq 2$. Then M contains an embedded superincompressible surface that is not a fiber in a fibration of M over S ¹ and that represents a nontrivial element of $H_2 (M, \partial M; Z)$. The Proof of this theorem will be finished in Section 2.3. Our proof is essentially the same as Thurston’s.