Orbifold Examples

Definition

Orbifolds are a generalization of manifolds that allow for certain types of singularities, typically arising from the quotient of a manifold by the action of a group. These structures are crucial in understanding geometrical and topological properties in noncommutative geometry, particularly when discussing spaces that exhibit symmetries and local singularities.

5 Must Know Facts For Your Next Test

1. Orbifolds can be constructed by taking a manifold and identifying points under the action of a group, creating singular points in the resulting structure.
2. Common examples of orbifolds include the cone over a circle and the disk with cone points, which exhibit specific types of singularities.
3. The classification of orbifolds can often be related to their underlying group actions and the nature of the singularities present.
4. In noncommutative geometry, orbifolds provide a way to study spaces with symmetries that may not be captured by traditional manifold theory.
5. Orbifolds can also arise in string theory and mirror symmetry, showcasing their importance in modern mathematical physics.

Review Questions

• How do orbifolds differ from traditional manifolds, particularly in terms of their geometric and topological properties?
  • Orbifolds differ from traditional manifolds primarily by allowing for singularities, which can arise from the quotienting process involving group actions. While manifolds are smooth and locally Euclidean everywhere, orbifolds can have points where local neighborhoods do not resemble Euclidean space due to these singularities. This difference is crucial for studying spaces with symmetries and understanding more complex geometrical structures.
• Discuss the role of group actions in the construction and classification of orbifolds.
  • Group actions are essential in defining orbifolds because they dictate how points in a manifold are identified to form the orbifold structure. The nature of these group actions determines the types of singularities that arise and influences the classification of the resulting orbifold. Understanding these actions allows mathematicians to categorize different orbifold types based on their symmetry properties and geometric characteristics.
• Evaluate how orbifold examples contribute to our understanding of noncommutative geometry and its applications in theoretical physics.
  • Orbifold examples significantly enhance our understanding of noncommutative geometry by illustrating how classical geometric concepts can be extended to include spaces with singularities and symmetries. They serve as models for spaces encountered in string theory and other areas of theoretical physics, where conventional manifolds may not suffice. By studying orbifolds, researchers can explore new mathematical frameworks that capture the behavior of physical systems at various scales, linking geometry, topology, and algebraic structures in innovative ways.

Related terms

Quotient Space: A quotient space is formed by partitioning a topological space into disjoint subsets, which are treated as single points in the new space.

Singularity: A singularity is a point at which a mathematical object is not well-behaved, for example, where it fails to be differentiable or where it ceases to be defined.

Group Action: A group action is a formal way of interpreting group elements as symmetries or transformations that can be applied to objects in a space.