Nonpositive Curvature and Reflection Groups
نویسندگان
چکیده
Introduction. A space is aspherical if its universal cover is contractible. Examples of aspherical spaces occur in differential geometry (as complete Riemannian manifolds of nonpositive sectional curvature), in Lie groups (as Γ\G/K where G is a Lie group, K is a maximal compact subgroup and Γ is a discrete torsion-free subgroup), in 3-manifold theory and as certain 2-dimensional cell complexes. The main purpose of this paper is to describe another interesting class of examples coming from the theory of reflection groups (also called “Coxeter groups”). One of the main results is explained in §3 and §9: given a finite simplicial complex L, there is a compact aspherical polyhedron X such that the link of each vertex in X is isomorphic to the barycentric subdivision of L. A version of this result first appeared in [D1]. Later, in [G], Gromov showed that the polyhedron X can be given a piecewise Euclidean metric which is nonpositively curved in the sense of Aleksandrov. This gives a new proof of the asphericity of X (cf. Theorem 1.5 below). As we vary the choice of the link L we get polyhedra X with a variety of interesting properties. For example, if L is homeomorphic to an (n− 1)-sphere, then X is an n-manifold. More examples are discussed in §11. Chapter I covers background material on nonpositive curvature. The main examples are discussed in Chapter II. Chapter III deals with some related aspherical complexes which arise in the study of complements of arrangements of hyperplanes. This paper began as a set of notes for three lectures which I gave at the Eleventh Annual Workshop in Geometric Topology at Park City, Utah in June 1994. In the course of preparing it for this volume I have added approximately 25 percent more material, notably, §4, §5, §10 and parts of §11. I would like to thank Lonette Stoddard for preparing the figures.
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تاریخ انتشار 2005