On the existence of infinitely many essential surfaces of bounded genus.
In [Topology 42, No. 4, 845--906 (2003; Zbl 1013.57013)], \textit{W. Jaco} and \textit{E. Sedgwick} used normal surface theory to prove the following theorem, which is a generalization of a theorem of Haken: Theorem 0.1 (Jaco-Sedgwick). Let \(M\) be an irreducible, boundary irreducible 3-manifold whose boundary is a single torus, and suppose that \(M\) contains no essential surfaces of genus 1, then \(M\) cannot contain infinitely many essential surfaces of uniformly bounded Euler characteristic. By essential we mean incompressible, boundary incompressible. The author of the paper under review uses branched surface theory to generalize this theorem, by allowing manifolds with arbitrary boundary. In the proof the author endows \(M\) with an arbitrary triangulation and isotopes all essential surfaces into normal surfaces with least area, where the area is defined to be the number of times the normal surface meets the 1-skeleton. An essential normal surface \(S\) is assigned an Euler ratio which is \(-\chi(S)/\text{Area}(S)\). The author then proves that for every manifold \(M\) as in the statement of the theorem there exists a lower bound \(K\) on the Euler ratios of all essential surfaces; this implies the main result. Note that this lower bound can be interpreted as ``large area connected essential surfaces have large genus''.
scientific article; zbMATH DE number 1818953