## Duke Mathematical Journal

### 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 $\tilde{M}$ satisfy Hruska’s isolated flats condition, and contain $2$-dimensional flats $F$ with the property that $\partial^{\infty}F\cong S^{1}\hookrightarrowS^{3}\cong\partial^{\infty}\tilde{M}$ are nontrivial knots. As a consequence, we obtain that the group $\pi_{1}(M)$ 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.

#### Article information

Source
Duke Math. J., Volume 161, Number 1 (2012), 1-28.

Dates
First available in Project Euclid: 30 December 2011

https://projecteuclid.org/euclid.dmj/1325264704

Digital Object Identifier
doi:10.1215/00127094-1507259

Mathematical Reviews number (MathSciNet)
MR2872552

Zentralblatt MATH identifier
1237.57015

#### Citation

Davis, M.; Januszkiewicz, T.; Lafont, J.-F. 4-dimensional locally CAT(0)-manifolds with no Riemannian smoothings. Duke Math. J. 161 (2012), no. 1, 1--28. doi:10.1215/00127094-1507259. https://projecteuclid.org/euclid.dmj/1325264704

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