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.
- Front Matter
- 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 Re°ection Groups Michael WDavis
- 1 Transformation groups and orbifolds
- 2 2-dimensional orbifolds
- 3 Re°ection groups
- 4 3-dimensional hyperbolic re°ection 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
- 4 The orbifold M1
- 5 Cubic curves and the universal curve E !M1
- 6 The Picard groups of M1;1 and M1
- 7 The algebraic topology of M1
- 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 ¯ber
- 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 Mari~no{Vafa formula of one-partition Hodge integrals
- 3 Applications of the Mari~no{Vafa formula
- 4 Three approaches to the Mari~no{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 LÄu
- 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 AWentworth
- 1 Introduction
- 2 Action of Out (¼) on representation varieties
- 3 Action on the cohomology of the space of °at unitary connections
- 4 Action on the cohomology of the SL (2
- References
