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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)
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[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
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