 
Summary: COMPLETIONS, BRANCHED COVERS, ARTIN
GROUPS AND SINGULARITY THEORY
DANIEL ALLCOCK
Abstract. We study the curvature of metric spaces and branched
covers of Riemannian manifolds, with applications in topology and
algebraic geometry. Here curvature bounds are expressed in terms
of the CAT() inequality. We prove a general CAT() extension
theorem, giving sufficient conditions on and near the boundary of a
locally CAT() metric space for the completion to be CAT(). We
use this to prove that a branched cover of a complete Riemannian
manifold is locally CAT() if and only if all tangent spaces are
CAT(0) and the base has sectional curvature bounded above by .
We also show that the branched cover is a geodesic space. Using
our curvature bound and a local asphericity assumption we give a
sufficient condition for the branched cover to be globally CAT()
and the complement of the branch locus to be contractible.
We conjecture that the universal branched cover of Cn
over
the mirrors of a finite Coxeter group is CAT(0). Conditionally
on this conjecture, we use our machinery to prove the Arnol
