MSJ Memoirs, Vol. 27. 171pp + xii.   Also available outside Japan at World Scientific Pub. Co. Download at Project Euclid.


This book exposes the connection between the low-dimensional orbifold theory and geometry that was first discovered by Thurston in 1970s
providing a key tool in his proof of the hyperbolization of Haken 3-manifolds. However, our main aim is to explain most of the topology of orbifolds
but to restrict our attention to the 2-dimensional orbifolds in application to the geometric structure theory.

The book was intended for the advanced undergraduates and the beginning graduate students who understand some differentiable manifold theory,
Riemannian geometry, some manifold topology, algebraic topology, and Lie group actions. But we do include sketches of these theories in
the beginning of the book as a review. Unfortunately, some familiarity with the category theory is needed where the author cannot provide
a sufficiently good introduction.

In this book, we tried to collect the theory of orbifolds scattered in various literatures for our purposes. Here, we set out to write down
the traditional approach to orbifolds using charts, and we include the categorical definition using groupoids. We think that understanding both theories
is necessary.

Computer experimentation is important in this field for understanding and research. We will also maintain a collection of Mathematica packages
we wrote at our homepages so that the readers can experiment with them. We will also give addresses where one can find the computational packages.
Many links will be gone soon enough but something related will only reappear in other places.

This book is based on courses that the author gave in the fall term of 2008 at Tokyo Institute of Technology as a visiting professor and the spring term
of 2011 at KAIST. I thank very much the hospitality of the Department of Mathematical and Computing Sciences at Tokyo Institute of Technology.
I thank David Fried, William Goldman, Craig Hodgson, Steve Kerckhoff, Hyuk Kim, Sadayoshi Kojima, Walter Neumann, Alan Reid, William Thurston,
and many others not mentioned here for introducing me the ideas of orbifolds and geometric structures on them. Some of the graphics were done with
Jfig© and Mathematica™ using the PoincareModel package from the Experimental Geometry Lab of the University of Maryland, College Park, written
by William Goldman available from http://egl.math.umd.edu/ and also the Curved Space by J. Weeks avaliable from
http://geometrygames.org/CurvedSpaces/index.html. We also thank Gye-Seon Lee and Kanghyun Choi for drawing many of the graphics using these packages.
I also thank many of my colleagues for much patience while this book was being written down. We also thank the anonymous referees for suggesting a number
of improvements. 

copyright: 2012 by the Mathematical Society of Japan

Downloads:

Table of Content: pdf
Project Euclid version
Version 3. July 29, 2012
Associated Mathematica files: zipped

Addendum

Errata