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Kneser-Haken finiteness for bounded 3-manifolds, locally free groups, and cyclic covers. (English) Zbl 0896.57012

The authors obtain the following main results:
Theorem 1. Let \(M\) be a compact 3-manifold with boundary and \(b\) an integer greater than zero. There is a constant \(c(M,b)\) so that if \(F_1,\dots,F_k\), \(k>c\), is a collection of incompressible surfaces such that all the Betti numbers \(b_1F_i< b\), \(1\leq i\leq k\), and no \(F_i\), \(1\leq i\leq k\), is a boundary parallel annulus or a boundary parallel disk, then at least two members \(F_i\) und \(F_j\) are parallel.
Theorem 2. Let \(N\) be an irreducible non-compact 3-manifold without boundary. The following conditions are equivalent:
(1) \(N\) contains no closed incompressible surface;
(2) \(N\) is exhausted by handlebodies, \(N= \bigcup_{i=1}^\infty H_i\), \(H_i\subset \text{interior } H_{i+1}\).
(3) \(\pi_1(N)\) is locally free.
Theorem 3. Let \(M\) be a compact manifold with each boundary component either a torus or a Klein bottle. Suppose \(\theta\in H^1(M,Z)\) has a nontrivial restriction to each boundary component. Then either (1) the infinite cyclic cover is homeomorphic to a compact surface \(F\times \text{Reals}\), \(M^\theta\cong F\times \mathbb{R}\), or (2) \(M^\theta\) contains a closed 2-sided incompressible surface \(\Sigma \subset M^\theta\). Thus, \(\pi_1(M^\theta)\) locally free implies \(\pi_1(M)\) free.

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
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