Summary: LOCAL-TO-ASYMPTOTIC 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 asymptotic-to-local results will be
difficult to achieve. Nevertheless, we are able to prove a partial converse
to our main local-to-asymptotic result.
Given a finite complex X, it would be useful to have a local condition that
implies that the universal cover X is n-acyclic or n-connected at infinity. For
example, in , 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 ). Let X be a finite, non-positively curved cubical
complex. If the link Lk(v) of each vertex v is n-acyclic, and the link remains
n-acyclic whenever you remove a simplex from Lk(v), then the universal
cover X is n-acyclic 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.