Duke Mathematical Journal

Completions, branched covers, Artin groups, and singularity theory

Daniel Allcock

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We study the curvature of metric spaces and branched covers of Riemannian manifolds, with applications in topology and algebraic geometry. Here curvature bounds are expressed in terms of the CAT ( χ ) inequality. We prove a general CAT ( χ ) extension theorem, giving sufficient conditions on and near the boundary of a locally CAT ( χ ) metric space for the completion to be CAT ( χ ) . We use this to prove that a branched cover of a complete Riemannian manifold is locally CAT ( χ ) if and only if all tangent spaces are CAT ( 0 ) and the base has sectional curvature bounded above by χ . We also show that the branched cover is a geodesic space. Using our curvature bound and a local asphericity assumption we give a sufficient condition for the branched cover to be globally CAT ( χ ) and the complement of the branch locus to be contractible.

We conjecture that the universal branched cover of C n over the mirrors of a finite Coxeter group is CAT ( 0 ) . This is closely related to a conjecture of Charney and Davis, and we combine their work with our machinery to show that our conjecture implies the Arnol$'$d–Pham–Thom conjecture on K ( π , 1 ) spaces for Artin groups. Also conditionally on our conjecture, we prove the asphericity of moduli spaces of amply lattice-polarized K3 surfaces and of the discriminant complements of all the unimodal hypersurface singularities in Arnol$'$d’s hierarchy.

Article information

Duke Math. J., Volume 162, Number 14 (2013), 2645-2689.

Received: 2 August 2012
Revised: 1 March 2013
First available in Project Euclid: 6 November 2013

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Zentralblatt MATH identifier

Primary: 51K10: Synthetic differential geometry
Secondary: 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 57N65: Algebraic topology of manifolds 20F36: Braid groups; Artin groups 14B07: Deformations of singularities [See also 14D15, 32S30]


Allcock, Daniel. Completions, branched covers, Artin groups, and singularity theory. Duke Math. J. 162 (2013), no. 14, 2645--2689. doi:10.1215/00127094-2380977. https://projecteuclid.org/euclid.dmj/1383760701

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