Geometric structures on 2-orbifolds. Exploration of discrete symmetry
type: Item
instance of: scholarly article
title: Geometric structures on 2-orbifolds. Exploration of discrete symmetry
zbMATH Open document ID: 1260.57043
author: Suhyoung Choi
published in: MSJ Memoirs
publication date: 2012-09-18T00:00:00Z
full work available at URL: http://projecteuclid.org/euclid.msjm/1418313992
reviewed by: Christopher Seaton
Mathematics Subject Classification ID: 57R18
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57S30
Mathematics Subject Classification ID: 20C99
zbMATH DE Number: 6083309
zbMATH Keywords: orbifold
zbMATH Keywords: 2-orbifold
zbMATH Keywords: geometric structure
zbMATH Keywords: deformation space
zbMATH Keywords: Teichmüller space
MaRDI profile type: Publication
OpenAlex ID: W630351897
DOI: 10.2969/MSJMEMOIRS/027010000
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review text: 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.
description: scientific article; zbMATH DE number 6083309