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.
KeywordsEuler Characteristic Homology Class Nontrivial Element Integer Coefficient Kernel Distribution
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