Duke Mathematical Journal

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

M. Davis, T. Januszkiewicz, and J.-F. Lafont

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We construct examples of 4-dimensional manifolds M supporting a locally CAT(0)-metric, whose universal covers satisfy Hruska’s isolated flats condition, and contain 2-dimensional flats F with the property that FS1S3 are nontrivial knots. As a consequence, we obtain that the group π1(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×K is a locally CAT(0)-manifold which does not support any Riemannian metric of nonpositive sectional curvature.

Article information

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

First available in Project Euclid: 30 December 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 20F67: Hyperbolic groups and nonpositively curved groups 20F55: Reflection and Coxeter groups [See also 22E40, 51F15]


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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