 
Summary: LOCALTOASYMPTOTIC TOPOLOGY
FOR COCOMPACT CAT(0) COMPLEXES
NOEL BRADY1, JON MCCAMMOND2, AND JOHN MEIER
Abstract. We give a local condition that implies connectivity at infin
ity properties for CAT(0) polyhedral complexes of constant curvature.
We show by various examples that asymptotictolocal results will be
difficult to achieve. Nevertheless, we are able to prove a partial converse
to our main localtoasymptotic result.
1. Introduction
Given a finite complex X, it would be useful to have a local condition that
implies that the universal cover X is nacyclic or nconnected at infinity. For
example, in [4], simple link conditions are given that ensure that a simply
connected, CAT(0) cubical complex has prescribed topological properties
at infinity. Namely,
Theorem 1.1 (4.1 in [4]). Let X be a finite, nonpositively curved cubical
complex. If the link Lk(v) of each vertex v is nacyclic, and the link remains
nacyclic whenever you remove a simplex from Lk(v), then the universal
cover X is nacyclic at infinity. Similarly, if Lk(v) is simply connected
and the link remains simply connected whenever you remove a simplex from
Lk(v), then the universal cover X is simply connected at infinity.
