PoincaréSphäre

The Poincaré sphere, also known as the Poincaré homaloid, is a topological space studied in mathematics, particularly in the field of topology and dynamical systems. It is a three-dimensional manifold that is homeomorphic to the three-sphere. It was introduced by the French mathematician Henri Poincaré.

The Poincaré sphere is characterized by its specific topological properties. It can be visualized as a solid sphere where the points on the boundary sphere are identified in a particular way. Specifically, antipodal points on the boundary are identified. This identification process results in a compact, connected, and orientable manifold without boundary.

A key feature of the Poincaré sphere is its relation to the Poincaré conjecture, which was one of the most famous unsolved problems in topology for many decades. The conjecture, now proven by Grigori Perelman, states that any simply connected, closed three-manifold is homeomorphic to the three-sphere. The Poincaré sphere serves as an example and a point of reference in discussions related to this conjecture and other problems in three-manifold theory. Its study has contributed significantly to the understanding of the fundamental shapes and structures of three-dimensional spaces.