4-dimensional locally CAT(0)-manifolds with no Riemannian smoothings

Abstract

We construct examples of 4-dimensional manifolds $M$ supporting a locally CAT(0)-metric, whose universal covers $\mathrm{M̃}$ satisfy Hruska’s isolated flats condition, and contain $2$-dimensional flats $F$ with the property that ${\partial }^{\infty }F\cong S^1\hookrightarrow S^3\cong {\partial }^{\infty }\mathrm{M̃}$ are nontrivial knots. As a consequence, we obtain that the group ${\pi }_1\left(M\right)$ cannot be isomorphic to the fundamental group of any compact Riemannian manifold of nonpositive sectional curvature. In particular, if $K$ is any compact locally CAT(0)-manifold, then $M\times K$ is a locally CAT(0)-manifold which does not support any Riemannian metric of nonpositive sectional curvature.