orbifold

orbifold. In mathematics, particularly within differential geometry and topology, an orbifold is a generalization of a manifold that allows for the presence of certain types of singularities. These singularities are locally modeled on the quotient of Euclidean space by the action of a finite group, such as a reflection group or a rotation group. The concept provides a powerful framework for studying spaces with symmetry and has become essential in areas ranging from geometric group theory to string theory.

Definition and basic concepts

The formal definition utilizes the language of atlases and charts, similar to that of a smooth manifold. A key difference is that each chart is a pair consisting of an open subset of Euclidean space and a finite group action by a subgroup of the general linear group. The transition maps between charts must be compatible with these local group actions. This structure encapsulates singular points, such as those arising from cone points or mirror boundaries, where the local geometry is not that of a standard vector space. Important related structures include the notions of an orbifold covering map and the stratification of an orbifold into its regular and singular loci.

Examples and constructions

A fundamental example is the quotient of the plane by a cyclic group of rotations, yielding a cone with a conical singularity. The moduli space of Riemann surfaces with marked points naturally has an orbifold structure due to the presence of automorphisms. Weighted projective space, studied in algebraic geometry, is a classic algebraic example. Common constructions include taking the quotient of a manifold by a properly discontinuous but not necessarily free action of a discrete group, such as the action of a crystallographic group on Euclidean space. The teardrop orbifold and the football orbifold are simple, low-dimensional examples with interesting geometric properties.

Orbifold fundamental group and covering spaces

The appropriate notion of a fundamental group for these spaces, often called the orbifold fundamental group, was defined by William Thurston. It incorporates information about the local singular structure and can be computed via van Kampen's theorem for orbifolds. The theory of covering spaces generalizes elegantly, with a universal covering orbifold that is a simply connected manifold if the original space is "good", or a manifold if it is "very good". This theory is deeply connected to the study of Fuchsian groups and Kleinian groups acting on the hyperbolic plane and hyperbolic 3-space.

Orbifold cohomology and characteristic classes

Several cohomology theories have been developed for orbifolds, aiming to capture both topological and geometric data. Chen-Ruan cohomology, introduced by Weimin Chen and Yongbin Ruan, is a landmark theory originating from string theory that adds a twisted sector to account for singularities. Orbifold de Rham cohomology can be defined using differential forms invariant under local group actions. Notions of characteristic classes, such as the orbifold Euler characteristic and Chern classes, are also well-defined. The Euler characteristic for orbifolds satisfies a generalization of the classical Riemann-Hurwitz formula.

Applications in geometry and physics

In geometry, orbifolds are central to the study of spherical space forms, flat manifolds, and the classification of 3-manifolds via Thurston's geometrization conjecture. They appear naturally as limits in the Gromov-Hausdorff convergence of sequences of Riemannian manifolds. In theoretical physics, particularly in string theory, orbifolds provide a crucial method for compactifying extra dimensions, leading to models with reduced supersymmetry. The AdS/CFT correspondence often involves orbifolds of anti-de Sitter space. Furthermore, they are used in the study of conformal field theory and in constructing new examples of Calabi-Yau manifolds.

Category:Differential geometry Category:Topology Category:Geometric group theory