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

TL;DR

This paper introduces a new obstruction to the existence of smooth Riemannian metrics with non-positive curvature on certain 4-manifolds, using topological link properties at infinity.

Contribution

It extends existing methods to construct examples of locally CAT(0) 4-manifolds with specific boundary link properties that prevent smooth Riemannian smoothing.

Findings

Constructed examples of 4-manifolds with non-trivial boundary links at infinity.

Demonstrated that these manifolds cannot admit smooth non-positively curved Riemannian metrics.

Showed all flats are unknotted at infinity, yet smoothing is impossible.

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.