# 2-Manifold Triangulations

## Chapter 4. 2-Manifold Triangulations

Regina also supports 2-manifold triangulations, which consist of a set of triangles with instructions on how some or all of their edges should be glued together in pairs.

2-manifold triangulations only play a minor role in Regina; it is with 3-manifold triangulations that the real action takes place. 2-manifold triangulations can (for example) arise from census generation, or by constructing vertex links from 3-manifold triangulations.

In the user interface, 2-manifold triangulations behave like a stripped-down version of 3-manifold triangulations. The interactions are much the same, and the reader is referred to the detailed chapter on 3-manifold triangulations for details.

## Identifying 2-Manifolds

One important difference between 2-manifolds and 3-manifolds is that the underlying 2-manifold is easy to recognise from its triangulation. For connected 2-manifold triangulations, Regina will display the exact 2-manifold at the top of the triangulation viewer, as illustrated below.

For orientable 2-manifolds, the genus refers to the number of handles. For non-orientable 2-manifolds, the genus refers to the number of crosscaps (so, for instance, the Klein bottle has genus two).

Regina will also display the Euler characteristic at the top of the triangulation viewer, marked with the symbol χ.