Essential disks and semi-essential surfaces in 3-manifolds (Q411822)

The present paper develops a new normal surface theory, where surfaces in general position with respect to the cellulation are ``carried'' by a finite set of spaces endowed with a smooth structure, called \textit{carriers}. Since the notion of carrier comes from performing surgery on a boundary compressing disk for a semi-essential surface (see Section 2 of the paper for details), the analysis is analogous to the analysis of incompressible surfaces using branched surfaces, see \textit{R. F. Williams} [Publ. Math., Inst. Hautes Étud. Sci. 43, 169--203 (1973; Zbl 0279.58013)] and \textit{L. Mosher} and \textit{U. Oertel} [Commun. Anal. Geom. 6, No. 1, 67--140 (1998; Zbl 0915.53020)]; moreover, the authors make use of a weaker version of the notion of \textit{train track}, see \textit{A. E. Hatcher} [Topology Appl. 30, No. 1, 63--88 (1988; Zbl 0662.57005)] and \textit{A. E. Hatcher} and \textit{U. Oertel} [Pac. J. Math. 154, No. 1, 87--101 (1992; Zbl 0772.57032)].NEWLINENEWLINEThe main theorem proves, for any compact orientable irreducible 3-manifold, the existence of finitely many carriers with some desired properties. The authors themselves point out the weakness of this result when compared with the main result of \textit{W. Floyd} and \textit{U. Oertel} [Topology 23, 117--125 (1984; Zbl 0524.57008)]. However, they hope the theory of carriers to be useful to describe objects representing limits of essential disks in a handlebody or in a manifold with compressible boundary: see, for example, \textit{H. Masur} [Ergodic Theory Dyn. Syst. 6, 99--116 (1986; Zbl 0628.57010)]; \textit{U. Oertel} [Topology 41, No. 2, 363--410 (2002; Zbl 0991.57017)] and \textit{L. N. Carvalho} [Geom. Topol. 10, 57--95 (2006; Zbl 1116.57017)].