Orbifolds and Stringy Topology

Abstract

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 product for orbifolds and has had significant impact. The final chapter includes explicit computations for a number of interesting examples.

Chen–Ruan Cohomology of \mathcal{M}_{1,n} and \overline{\mathcal{M}}_{1,n}

“…The latter framework has produced important results in algebraic geometry, topology, mathematical and theoretical physics, representation theory and number theory. On the other hand, our results are the first steps towards the study and the understanding of more recent topics of investigation, such as the stringy topology and the Gromov-Witten theory of the moduli spaces of curves (see, for example, [3]). …”

The Frölicher-type inequalities of foliations

“…[1] for basic definitions and properties of orbifolds). Furthermore every orbifold can be realized as a space of leaves of a Riemannian foliation (see [20] and [16]).…”

Nonvanishing vector fields on orbifolds

“…In this case, the obstruction was an element of the Chen-Ruan orbifold cohomology (see [4] or [1]), which is additively the cohomology of the inertia orbifold. For cyclic orbifolds, the cohomology of the inertia orbifold is large enough to produce a complete obstruction.…”

“…an orbifold that admits a presentation as M/G where M is a smooth manifold and G is a finite group acting smoothly. Using the techniques of [4] and [1], we produce a similar construction for orbifolds that do not admit such a presentation.…”

“…However, the construction contained here relates specifically to and generalizes existing constructions for quotients, including orbifold Euler characteristics. In a forthcoming paper, the authors will explore the relationship between the Γ-sectors given here and other constructions, including the inertia orbifold, the space of multi-sectors in [4] and [1], and orbifold Euler characteristics. Here, our focus is on the properties of the construction itself and the obstruction to nonvanishing vector fields.…”