Covering spaces of 3-manifolds. (English) Zbl 0624.57016

Let M be an arbitrary closed \(P^ 2\)-irreducible 3-manifold. The authors announce, and outline the proofs, of the following two theorems. If \(\pi_ 1(M)\) contains the fundamental group of a closed surface other than \(S^ 2\) or \(P^ 2\), then the universal covering of M is \({\mathbb{R}}^ 3\). If \(\pi_ 1(M)\) contains \({\mathbb{Z}} \times {\mathbb{Z}}\), the every covering of M with fundamental group \({\mathbb{Z}} \times {\mathbb{Z}}\) can be compactified by adding boundary to it.
Reviewer: J.Vrabec


57N10 Topology of general \(3\)-manifolds (MSC2010)
57M10 Covering spaces and low-dimensional topology
Full Text: DOI


[1] Francis Bonahon, Bouts des variétés hyperboliques de dimension 3, Ann. of Math. (2) 124 (1986), no. 1, 71 – 158 (French). · Zbl 0671.57008
[2] Michael Freedman, Joel Hass, and Peter Scott, Least area incompressible surfaces in 3-manifolds, Invent. Math. 71 (1983), no. 3, 609 – 642. · Zbl 0482.53045
[3] William Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, R.I., 1980. · Zbl 0433.57001
[4] W. Jaco and H. Rubinstein, P-L minimal surfaces in 3-manifolds, preprint. · Zbl 0652.57005
[5] Peter Scott, A new proof of the annulus and torus theorems, Amer. J. Math. 102 (1980), no. 2, 241 – 277. · Zbl 0439.57004
[6] Jonathan Simon, Compactification of covering spaces of compact 3-manifolds, Michigan Math. J. 23 (1976), no. 3, 245 – 256 (1977). · Zbl 0331.57002
[7] R. Schoen and Shing Tung Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2) 110 (1979), no. 1, 127 – 142. · Zbl 0431.53051
[8] W. Thurston, Geometry and topology of 3-manifolds, Lecture Notes, Princeton University, 1978-1979.
[9] 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.