# zbMATH — the first resource for mathematics

Essential closed surfaces in bounded 3-manifolds. (English) Zbl 0896.57009
An essential closed surface $$S$$ in a 3-manifold $$M$$ is an immersion of $$S$$ into $$M$$ such that the image of $$S$$ cannot be freely homotoped into $$\partial M$$. The authors answer the important question of whether a given compact irreducible 3-manifold $$M$$ contains essential closed surfaces. They show that if $$M$$ has non-empty incompressible boundary then either $$M$$ is covered by a product (closed surface)$$\times I$$, or $$M$$ contains an essential closed surface $$S$$. This implies in particular that any complement of a knot in $$S^3$$ contains an essential closed surface. In the proof the authors show that there is an embedding of an incompressible closed surface in an infinite cyclic cover of a finite cover of M and this yields an other interesting Theorem: Either $$\partial M$$ consists only of tori and M is covered by a product $$T^2\times I$$, or $$\pi_1(M)$$ virtually maps onto a free group of rank 2. As a further application of the proof the authors show that one can create nonperipheral 2-dimensional homology by passing to a finite sheeted covering of $$M$$.

##### MSC:
 57N10 Topology of general $$3$$-manifolds (MSC2010)
##### Keywords:
essential surface; 3-manifold; knot complement
Full Text:
##### References:
 [1] J. W. Anderson, Closed essential surfaces in hyperbolizable acylindrical manifolds. Preprint. · Zbl 0902.57020 [2] B. Freedman & M.H. Freedman Haken finiteness for bounded $$3$$-manifolds, locally free groups and cyclic covers. Preprint. · Zbl 0896.57012 [3] David Gabai, On 3-manifolds finitely covered by surface bundles, Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 112, Cambridge Univ. Press, Cambridge, 1986, pp. 145 – 155. [4] John Hempel, Residual finiteness for 3-manifolds, Combinatorial group theory and topology (Alta, Utah, 1984) Ann. of Math. Stud., vol. 111, Princeton Univ. Press, Princeton, NJ, 1987, pp. 379 – 396. · Zbl 0772.57002 [5] John Hempel, 3-Manifolds, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. · Zbl 0345.57001 [6] D. D. Long and G. A. Niblo, Subgroup separability and 3-manifold groups, Math. Z. 207 (1991), no. 2, 209 – 215. · Zbl 0711.57002 [7] Martin Scharlemann and Ying Qing Wu, Hyperbolic manifolds and degenerating handle additions, J. Austral. Math. Soc. Ser. A 55 (1993), no. 1, 72 – 89. · Zbl 0802.57005 [8] W.P. Thurston. The Geometry and Topology of 3-manifolds. Princeton University mimeographed notes. (1979) [9] William P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357 – 381. · Zbl 0496.57005 [10] Friedhelm Waldhausen, On irreducible 3-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56 – 88. · Zbl 0157.30603
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.