Fundamental Groups of Blowups
Abstract
Many examples of nonpositively curved closed manifolds arise as blowups of projective hyperplane arrangements. If the hyperplane arrangement is associated to a finite reflection group W, and the blowup locus is Winvariant, then the resulting manifold M will admit a cell decomposition whose maximal cells are all combinatorially isomorphic to a given convex polytope P. In other words, M admits a tiling with tile P. The universal covers of such examples yield tilings of R^n whose symmetry groups are generated by involutions but are not, in general, reflection groups. We begin a study of these ``mock reflection groups'', and develop a theory of tilings that includes the examples coming from blowups and that generalizes the corresponding theory of reflection tilings. We apply our general theory to classify the examples coming from blowups in the case where the tile P is either the permutohedron or the associahedron.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 March 2002
 arXiv:
 arXiv:math/0203127
 Bibcode:
 2002math......3127D
 Keywords:

 Geometric Topology