Topology: Soham's Ball

Soham's Ball

Soham's ball is a concept in mathematics, specifically in the field of topology.

Definition

Soham's ball is a topological space that is defined as the quotient space of a 3-ball (a three-dimensional ball) by identifying the northern hemisphere with the southern hemisphere in a certain way.

Properties

• Soham's ball is a non-manifold, meaning it is not a topological manifold.
• It is a compact, connected, and simply-connected space.
• Soham's ball is not a triangulable space, meaning it cannot be subdivided into a finite number of simplices (i.e., points, lines, triangles, etc.).

Significance

Soham's ball is an important example in topology because it shows that not all compact, connected, and simply-connected spaces are manifolds. It is often used as a counterexample to illustrate the limitations of certain topological properties.

Soham's Ball

• A topological space in mathematics, specifically in the field of topology.

Definition

• Soham's ball is a quotient space of a 3-ball (a three-dimensional ball).
• The northern hemisphere is identified with the southern hemisphere in a certain way.

Properties

• Non-manifold, meaning it is not a topological manifold.
• Compact, connected, and simply-connected space.
• Not a triangulable space, meaning it cannot be subdivided into a finite number of simplices (e.g., points, lines, triangles, etc.).

Significance

• Important example in topology, showing that not all compact, connected, and simply-connected spaces are manifolds.
• Often used as a counterexample to illustrate the limitations of certain topological properties.