爆破的基本群
Fundamental Groups of Blow-ups
摘要：
许多非正曲闭流形的例子都是由于射影超平面排列的爆破而产生的。如果超平面排列与有限反射群W相关联，并且爆破轨迹是W-不变的，则所得到的流形M将允许一个胞腔剖分，其最大细胞都组合同构于给定的凸多胞体P。换句话说，M允许一个具有瓦片P的瓦片。这样的例子的泛覆盖产生R^n的瓦片，其对称群由对合生成，但不是，一般来说，反射组。我们开始研究这些“模拟反射群”，并开发出一个理论的瓷砖，其中包括来自爆破的例子，并概括了相应的理论反射瓷砖。我们应用我们的一般理论来分类的情况下，瓷砖P是permutohedron或associahedron来自爆破的例子。
英文摘要：
Many examples of nonpositively curved closed manifolds arise as blow-ups of projective hyperplane arrangements. If the hyperplane arrangement is associated to a finite reflection group W, and the blow-up locus is W-invariant, 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 blow-ups and that generalizes the corresponding theory of reflection tilings. We apply our general theory to classify the examples coming from blow-ups in the case where the tile P is either the permutohedron or the associahedron.
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