Group-theoretic conditions under which closed aspherical manifolds are covered by Euclidean space.

*(English)*Zbl 1055.57032A 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.

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.

Reviewer: Bruce Hughes (Nashville)

##### 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 |