2015•arXiv (Cornell University)Open access
Hyperbolicity of contractible manifolds
Karim Adiprasito, Louis Funar
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Abstract
Addressing a question going back to Gromov, we show that, for an open, contractible manifold M of dimension at least 6, the following conditions are equivalent:
- M admits a geodesically complete CAT(-1) metric;
- M admits a geodesically complete CAT(0) metric;
- M is pseudo-collarable;
- M can be collapsed.
By work of Guilbault, this implies that the CAT(-1) property of a contractible manifold is purely determined by the structure at infinity. It also yields many new contractible manifolds that admit CAT(-1) metrics, and demonstrates that contractible manifolds of the Whitehead type can never admit a complete CAT(-1) metric.