Groups generated by reflections and aspherical manifolds not covered by Euclidean space. (English) Zbl 0531.57041

Davis, Michael W.

Ann. Math. (2) 117, 293-324 (1983).

The paper presents a very interesting new construction of proper actions of Coxeter groups on contractible manifolds. This construction is applied to give classes of counterexamples in every dimension \(\geq 4\) to the following conjecture. The universal cover of any closed aspherical manifold is homeomorphic to Euclidean space [F. E. A. Johnson, Proc. Camb. Philos. Soc. 75, 165-173 (1974; Zbl 0278.57006)]. So there are in every dimension \(\geq 4\) contractible manifolds not homeomorphic to Euclidean space admitting a proper free action of some discrete group with compact orbit space.

Reviewer: H.Abels

MSC:

57S30	Discontinuous groups of transformations
57N15	Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
54H15	Transformation groups and semigroups (topological aspects)

Keywords:

groups generated by reflections; universal cover of closed aspherical manifold; proper actions of Coxeter groups on contractible manifolds

Citations:

Zbl 0278.57006