On the fundamental groups of trees of manifolds

Abstract

We consider limits of inverse sequences of closed manifolds, whose consecutive terms are obtained by connect summing with closed manifolds, which are in turn trivialized by the bonding maps. Such spaces, which we refer to as trees of manifolds, need not be semilocally simply connected at any point and can have complicated fundamental groups. Trees of manifolds occur naturally as visual boundaries of standard nonpositively curved geodesic spaces, which are acted upon by right-angled Coxeter groups whose nerves are closed PL-manifolds. This includes, for example, those Coxeter groups that act on Davis' exotic open contractible manifolds. Also, all of the homogeneous cohomology manifolds constructed by Jakobsche are trees of manifolds. In fact, trees of manifolds of this type, when constructed from PL-homology spheres of common dimension at least 4, are boundaries of negatively curved geodesic spaces. We prove that if Z is a tree of manifolds, the natural homomorphism φ: π 1 (Z) → π 1 (Z) from its fundamental group to its first shape homotopy group is injective. If Z = bdy X is the visual boundary of a nonpositively curved geodesic space X, or more generally, if Z is a Z-set boundary of any ANR X, then the first shape homotopy group of Z coincides with the fundamental group at infinity of X: π 1 (Z) = π 1 ∞ (X). We therefore obtain an injective homomorphism ψ: π 1 (Z) → π 1 ∞ (X), which allows us to study the relationship between these groups. In particular, if Z = bdy r is the boundary of one of the Coxeter groups Γ mentioned above, we get an injective homomorphism ψ: π 1 (bdy F) → π 1 ∞ (Γ).