Link Obstruction to Riemannian smoothings of locally CAT(0) 4-manifolds

Abstract

We extend the methods of Davis-Januszkiewicz-Lafont to provide a new obstruction to smooth Riemannian metric with non-positive sectional curvature. We construct examples of locally CAT(0) 4-manifolds $M$, whose universal covers satisfy isolated flats condition and contain 2-dimensional flats with the property that $\sqcup\ \partial^\infty F_i \hookrightarrow \partial^\infty \tilde M$ are non-trivial links that are not isotopic to any great circle link. Further, all the flats in $\tilde M$ are unknotted at infinity, and yet $M$ does not have a Riemannian smoothing.