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**Orbifolds and stringy topology.**
*(English)*
Zbl 1157.57001

Cambridge Tracts in Mathematics 171. Cambridge: Cambridge University Press (ISBN 978-0-521-87004-7/hbk). xii, 149 p. (2007).

The book under review is an excellent introduction to the recent and interesting developements in geometry and topology of orbifolds motivated by string theory. In particular it gives a nice account of Chen-Ruan cohomology accessible to students.

The first three chapters are devoted to the classical geometric and topological notions of orbifolds. The basic definitions and constructions include orbifolds in terms of groupoids, effective orbifolds and orbifold fundamental group, amongst others. Special care is devoted to the notion of orbifold morphism, as well as inertia orbifolds, which play a central role in Chen-Ruan cohomology. De Rham cohomology, bundle theory and \(K\)-theory and effective orbifolds are also treated in the first part of the book, together with many examples and applications, especially in the context of complex algebraic geometry and resolution of singularities.

Chapter 4 is the core of the book and it is devoted to Chen-Ruan cohomology [W. Chen and Y. Ruan, Commun. Math. Phys. 248, No. 1, 1–31 (2004; Zbl 1063.53091)], using the tools of the first part. This includes the cup product which is new in both mathematics and physics, and can be viewed as a classical limit of Chen-Ruan quantum product. Cohomology twisted by a discrete group is also considered. Finally, Chapter 5 is devoted to computation of examples: abelian orbifolds and symmetric products.

The first three chapters are devoted to the classical geometric and topological notions of orbifolds. The basic definitions and constructions include orbifolds in terms of groupoids, effective orbifolds and orbifold fundamental group, amongst others. Special care is devoted to the notion of orbifold morphism, as well as inertia orbifolds, which play a central role in Chen-Ruan cohomology. De Rham cohomology, bundle theory and \(K\)-theory and effective orbifolds are also treated in the first part of the book, together with many examples and applications, especially in the context of complex algebraic geometry and resolution of singularities.

Chapter 4 is the core of the book and it is devoted to Chen-Ruan cohomology [W. Chen and Y. Ruan, Commun. Math. Phys. 248, No. 1, 1–31 (2004; Zbl 1063.53091)], using the tools of the first part. This includes the cup product which is new in both mathematics and physics, and can be viewed as a classical limit of Chen-Ruan quantum product. Cohomology twisted by a discrete group is also considered. Finally, Chapter 5 is devoted to computation of examples: abelian orbifolds and symmetric products.

Reviewer: Joan Porti (Bellaterra)

### MSC:

57-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes |

57R19 | Algebraic topology on manifolds and differential topology |

53D45 | Gromov-Witten invariants, quantum cohomology, Frobenius manifolds |

55N35 | Other homology theories in algebraic topology |

19L64 | Geometric applications of topological \(K\)-theory |