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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 \(M\).

57N10 Topology of general \(3\)-manifolds (MSC2010)
Full Text: DOI
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