Geometric structures on 2-orbifolds. Exploration of discrete symmetry
This book is an introduction to orbifolds and geometric structures of \(2\)-dimensional orbifolds. As the author states in the introduction, it is intended for advanced undergraduate or beginning graduate students with some background in topology, manifolds, differential geometry, and category theory. After reviewing prerequisite material on manifolds, Lie groups, compact Lie group actions, geometry, and discrete group actions in Chapters 2 and 3, Chapter 4 gives an introduction to the topology of orbifolds. This includes bundles over orbifolds, triangulations, covering spaces, and the orbifold fundamental group; most of this material is presented for orbifolds of arbitrary dimension. The definition of orbifold presented is that most common in low-dimensional topology: effective orbifolds using orbifold charts, though the author also introduces the alternative formulation as a Morita equivalence class of proper, étale, Lie groupoids. Chapter 6 introduces geometric structures on orbifolds and deformation spaces of geometric structures.NEWLINENEWLINEIn Chapters 5, 7, and 8, \(2\)-dimensional orbifolds and their geometric structures are considered. Chapter 5 discusses the topology of \(2\)-orbifolds including the classification up to diffeomorphism of \(2\)-orbifolds and \(1\)-dimensional suborbifolds, the orbifold Euler characteristic, and constructions of \(2\)-orbifolds using splitting and sewing. Chapter 7 covers deformation spaces of hyperbolic structures (Teichmüller spaces) of \(2\)-orbifolds using the decomposition of \(2\)-orbifolds into elementary \(2\)-orbifolds. Chapter 8 presents deformation spaces of projective structures on \(2\)-orbifolds.
scientific article; zbMATH DE number 6083309
scientific article