Introduction to Orbifold Theory
The notion of an orbifold is a generalization of the notion of a manifold which appears naturally in the context of properly discontinuous nonfree actions of groups on manifolds. Orbifolds were first invented by Satake [Sat57] in the 1950s under the name of V-manifolds. They were reinvented under the name of orbifolds by Thurston in the 1970s as a technical tool for proving the Hyperbolization Theorem. Our discussion of orbifolds follows mainly [Thu81, Chapter 13] and [Sco83a]. For a more detailed discussion of foundational issues of orbifold theory, we refer the reader to [Rat94, Chapter 13]. Note that many definitions that we take for granted in manifold theory are rather mysterious in the category of orbifolds. For instance, it is unclear what the correct definition of characteristic classes of orbifolds is (compare [Kaw78] and [HH90]). Even if one can define the Euler characteristic of an orbifold, it is unclear what the Betti numbers are. We will define the fundamental group of an orbifold, but it is far from obvious what the correct definition of higher homotopy groups, etc., is. All in all, it seems that orbifolds live in the grey area between the “good old manifold kingdom” and the dark world populated by the “wild creatures” of Alain Connes’ noncommutative geometry.
KeywordsFundamental Group Euler Characteristic Singular Locus Solid Torus Black Vertex
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