Does every surface admit a quadrangulation?

Question

Every surface (2-manifold) admits a triangulation, and I wonder if the same can be said for quadrangulation.

My intuition is that every orientable surface can be quadrangulated, but I'm not sure about non-orientable surfaces. Is there any theorem in this regard?

Answer 1

It seems to me that you can convert any triangulation into a quadrangulation. Let $x_1$, $x_2$, $x_3$, ... be the vertices of the triangulation. For each edge $(x_i, x_j)$ of the triangulation, let $y_{ij}$ be a point on the edge $(x_i, x_j)$. For each triangle $(x_i, x_j, x_k)$, let $z_{ijk}$ be a point in the interior of the triangle.

Then replace each triangle $(x_i, x_j, x_k)$ with the three quadrilaterals $(x_i, y_{ij}, z_{ijk}, y_{ik})$, $(x_j, y_{jk}, z_{ijk}, y_{ij})$ and $(x_k, y_{ik}, z_{ijk}, y_{jk})$.

Here is a picture: