An introduction to the theory of orbifolds from a modern perspective, combining techniques from geometry, algebraic topology and algebraic geometry. One of the main motivations, and a major source of examples, is string theory, where orbifolds play an important role. The subject is first developed following the classical description analogous to manifold theory, after which the book branches out to include the useful description of orbifolds provided by groupoids, as well as many examples in the context of algebraic geometry. Classical invariants such as de Rham cohomology and bundle theory are developed, a careful study of orbifold morphisms is provided, and the topic of orbifold K-theory is covered. The heart of this book, however, is a detailed description of the Chen-Ruan cohomology, which introduces a new product for orbifolds and has had significant impact in recent years. The final chapter includes explicit computations for a number of interesting examples.
Author(s): Alejandro Adem, Johann Leida, Yongbin Ruan
Series: Cambridge Tracts in Mathematics
Publisher: Cambridge University Press
Year: 2007
Language: English
Commentary: 25236
Pages: 164
Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 7
Introduction......Page 9
1.1 Classical effective orbifolds......Page 15
1.2 Examples......Page 19
1.3 Comparing orbifolds to manifolds......Page 24
1.4 Groupoids......Page 29
1.5 Orbifolds as singular spaces......Page 42
2.1 De Rham and singular cohomology of orbifolds......Page 46
2.2 The orbifold fundamental group and covering spaces......Page 53
2.3 Orbifold vector bundles and principal bundles......Page 58
2.4 Orbifold morphisms......Page 61
2.5 Classification of orbifold morphisms......Page 64
3.1 Introduction......Page 70
3.2 Orbifolds, group actions, and Bredon cohomology......Page 71
3.3 Orbifold bundles and equivariant K-theory......Page 74
3.4 A decomposition for orbifold K-theory......Page 77
3.5 Projective representations, twisted group algebras, and extensions......Page 83
3.6 Twisted equivariant K-theory......Page 86
3.7 Twisted orbifold K-theory and twisted Bredon cohomology......Page 90
4 Chen–Ruan cohomology......Page 92
4.1 Twisted sectors......Page 94
4.2 Degree shifting and Poincare pairing......Page 98
4.3 Cup product......Page 102
4.4 Some elementary examples......Page 109
4.5 Chen–Ruan cohomology twisted by a discrete torsion......Page 112
5.1 Abelian orbifolds......Page 119
5.1.1 The de Rham model......Page 120
5.1.2 Examples......Page 125
5.2 Symmetric products......Page 129
5.2.1 The Heisenberg algebra action......Page 130
5.2.2 The obstruction bundle......Page 142
5.2.3 LLQW axioms......Page 145
References......Page 152
Index......Page 160
