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Group-theoretic conditions under which closed aspherical manifolds are covered by Euclidean space. (English) Zbl 1055.57032
A well-known problem is to decide which closed aspherical \(m\)-manifolds \(M\) have universal cover \(\widetilde M\) homeomorphic to Euclidean space. It is known that if \(m\geq 3\) (and \(M\) is irreducible if \(m=3\)), then \(\widetilde M\) is homeomorphic to Euclidean space if and only if \(\widetilde M\) is simply connected at infinity, which is a geometric property of \(\pi_1M\). Hence, it is important to know which finitely presented groups are simply connected at infinity.
Two of the main results of the paper under review are a new geometric proof of a theorem of C. Houghton [J. Lond. Math. Soc., II. Ser. 15, 465–471 (1977; Zbl 0354.20025)] and B. Jackson [Topology 21, 71–81 (1981; Zbl 0472.57001)], namely, if \(1\to H\to G\to Q\to 1\) is a short exact sequence of finitely presented infinite groups and either \(H\) or \(Q\) is one-ended, then \(G\) is simply connected at infinity, and a generalization of it obtained by relaxing the normality condition on \(H\) in \(G\).
The authors also give a geometric proof of the theorem: {If \(M\) is a closed aspherical \(m\)-manifold (and irreducible if \(m=3\)) and \(\pi_1M\) contains a non-trivial cyclic normal subgroup, then \(\widetilde{M}\) is homeomorphic to Euclidean space.}
For \(m\geq 5\) this is due to R. Lee and F. Raymond [Topology 14, 49–57 (1975; Zbl 0313.57005)] and for \(m=3\) it follows from results of D. Gabai [Ann. Math. (2) 136, No. 3, 447–510 (1992; Zbl 0785.57004)] and A. Casson and D. Jungreis [Invent. Math. 118, No. 3, 441–456 (1994; Zbl 0840.57005)]. M. Davis [Ann. Math. (2) 117, 293–324 (1983; Zbl 0531.57041)] constructed the first examples of closed aspherical manifolds not covered by Euclidean space and the authors analyze those examples in the light of their results.

MSC:
57N99 Topological manifolds
57N10 Topology of general \(3\)-manifolds (MSC2010)
57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
57S30 Discontinuous groups of transformations
57M07 Topological methods in group theory
57M10 Covering spaces and low-dimensional topology
20F65 Geometric group theory
20F69 Asymptotic properties of groups
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