变换群与曲线模空间

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

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.

《变换群与曲线模空间》是由高等教育出版社出版的。