an:02092542
Zbl 1055.57032
Fischer, Hanspeter; Wright, David G.
Group-theoretic conditions under which closed aspherical manifolds are covered by Euclidean space
EN
Fundam. Math. 179, No. 3, 267-282 (2003).
0016-2736 1730-6329
2003
j
57N99 57N10 57N15 57S30 57M07 57M10 20F65 20F69
aspherical manifolds; universal cover; simply connected at infinity; ends of groups
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 \textit{C. Houghton} [J. Lond. Math. Soc., II. Ser. 15, 465--471 (1977; Zbl 0354.20025)] and \textit{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 \textit{R. Lee} and \textit{F. Raymond} [Topology 14, 49--57 (1975; Zbl 0313.57005)] and for \(m=3\) it follows from results of \textit{D. Gabai} [Ann. Math. (2) 136, No. 3, 447--510 (1992; Zbl 0785.57004)] and \textit{A. Casson} and \textit{D. Jungreis} [Invent. Math. 118, No. 3, 441--456 (1994; Zbl 0840.57005)]. \textit{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.
Bruce Hughes (Nashville)
0354.20025; 0472.57001; 0313.57005; 0785.57004; 0840.57005; 0531.57041