变换群在微分几何、几何拓扑、代数拓扑、代数几何、数论等诸多数学领域起到了基础性的作用。本书包含了2008年两个暑期学校“Transformation Groups and Orbifolds”和“Geometry of Teichmüller Spaces and Moduli Spaces of Curves”的扩展讲义，是相关专业学生和研究人员学习变换群、轨形、Teichmüller空间、映射类群、曲线模空间和相关课题的颇具价值的资料。本书可供数学专业的研究生和高年级本科生阅读，也可供相关领域研究人员参考。

Transformation groups have played a fundamental role in many areas of mathematics such as differential geometry, geometric topology, algebraic topology, algebraic geometry, number theory. Ore of the basic reasons for their importance is that symmetries are described by groups (or rather group actions). Quotients of smooth manifolds by group actions are usually not smooth manifolds. On the other hand, if the actions of the groups are proper, then the quotients are orbifolds. An important example is given by the action of the mapping class groups on the Teichmuller spaces, and the quotients give the moduli spaces of Riemann surfaces (or algebraic curves) and are orbifolds.

This book consists of expanded lecture' notes of two summer schools Transformation Groups and Orbifolds and Geometry of Teichmuller Spaces and Moduli Spaces of Curves in 2008 and will be a valuable source for people to learn transformation groups, orbifolds, Teichmuller spaces, mapping class groups, moduli soaces of curves and related topics.

Lectures on Orbifolds and Group Cohomology

 Alejandro Adem and Michele Klaus

 1 Introduction

 2 Classical orbifolds

 3 Examples of orbifolds

 4 Orbifolds and manifolds

 5 Orbifolds and groupoids

 6 The orbifold Euler characteristic and K-theory

 7 Stringy products in K-theory

 8 Twisted version

 References

Lectures on the Mapping Class Group of a Surface

 Thomas Kwok-Keung Au, Feng Luo and Tian Yang

 Introduction

 1 Mapping class group

 2 Dehn-Lickorish Theorem

 3 Hyperbolic plane and hyperbolic surfaces

 4 Quasi-isometry and large scale geometry

 5 Dehn-Nielsen Theorem

 References

Lectures on Orbifolds and Reflection Groups

 Michael W. Davis

 1 Transformation groups and orbifolds

 2 2-dimensional orbifolds

 3 Reflection groups

 4 3-dimensional hyperbolic reflection groups

 5 Aspherical orbifolds

 References

Lectures on Moduli Spaces of Elliptic Curves

 Richard Hain

 1 Introduction to elliptic curves and the moduli problem

 2 Families of elliptic curves and the universal curve

 3 The orbifold M1,1

 4 The orbifold ■1,1 and modular forms

 5 Cubic curves and the universal curve ■→■1,1

 6 The Picard groups of M1,1 and ■1,1

 7 The algebraic topology of ■1,1

 8 Concluding remarks

 Appendix A Background on Riemann surfaces

 Appendix B A very brief introduction to stacks

 References

An Invitation to the Local Structures of Moduli of Genus One Stable Maps

 Yi HU

 1 Introduction

 2 The structures of the direct image sheaf

 3 Extensions of sections on the central fiber

 References

Lectures on the ELSV Formula

 Chiu-Chu Melissa Liu

 1 Introduction

 2 Hurwitz numbers and Hodge integrals

 3 Equivariant cohomology and localization

 4 Proof of the ELSV formula by virtual localization

 References

Formulae of One-partition and Two-partition Hodge Integrals

 Chiu-Chu Melissa Liu

 1 Introduction

 2 The Marino-Vafa formula of one-partition Hodge integrals

 3 Applications of the Marifio-Vafa formula

 4 Three approaches to the Marino-Vafa formula

 5 Proof of Proposition 4.3

 6 Generalization to the two-partition case

 References

Lectures on Elements of Transformation Groups and Orbifolds

 Zhi Lu

 1 Topological groups and Lie groups

 2 G-actions (or transformation groups) on topological spaces

 3 Orbifolds

 4 Homogeneous spaces and orbit types

 5 Twisted product and slice

 6 Equivariant cohomology

 7 Davis-Januszkiewicz theory

 References

The Action of the Mapping Class Group on Representation Varieties

 Richard A. Wentworth

 1 Introduction

 2 Action of Out (π) on representation varieties

 3 Action on the cohomology of the space of fiat unitary connections

 4 Action on the cohomology of the SL (2, C) character variety

References