halfsurface

A halfsurface is a mathematical concept that refers to a surface that is locally two-dimensional but globally one-dimensional. It is a type of topological space that can be visualized as a surface that has been cut along a curve, resulting in a space that is not simply connected. The term "halfsurface" is often used in the context of low-dimensional topology and geometric group theory.

In more concrete terms, a halfsurface can be thought of as a surface with a boundary, where the boundary is a single simple closed curve. This curve divides the surface into two regions: the "inside" and the "outside." The halfsurface is the union of the boundary curve and the inside region. The outside region is not part of the halfsurface, but it is often considered in the study of halfsurfaces because it can influence the topological properties of the halfsurface.

Halfsurfaces can be characterized by their genus, which is a topological invariant that measures the number of "holes" in the surface. A halfsurface of genus zero is a disk, while a halfsurface of genus one is an annulus. Higher genus halfsurfaces can be constructed by attaching handles to the surface.

The study of halfsurfaces is important in various areas of mathematics, including the classification of 3-manifolds, the study of mapping class groups, and the theory of Teichmüller spaces. In particular, halfsurfaces play a key role in the proof of the geometrization conjecture, a major result in low-dimensional topology.