Compact non-orientable surface

This article is about compact non-orientable surfaces, i.e., two-dimensional compact connected non-orientable manifolds.

Classification
There is a bijection:

Positive integers $$\leftrightarrow$$ Homeomorphism classes of compact non-orientable surfaces

The correspondence, in the forward direction, is as follows: given a positive integer $$k$$, the corresponding compact non-orientable surface is a connected sum of $$k$$ copies of the real projective plane.

If we denote by $$P$$ the real projective plane, then we have that $$P \# P$$ is the Klein bottle, which we denote by $$K$$, and that $$P\# K \cong P \# T$$ where $$T$$ is the 2-torus (which is orientable). $$P \# K$$ is termed Dyck's surface and the fact that it is homeomorphic to $$P \# T$$ is termed Dyck's theorem.

Using this and some further manipulation, we can conclude that:


 * For odd $$k$$, the $$k$$-fold connected sum of $$P$$ with itself can be identified with the connected sum of $$P$$ and $$(k-1)/2$$ copies of the 2-torus.
 * For even $$k$$, the $$k$$-fold connected sum of $$P$$ with itself can be identified with the connected sum of the Klein bottle $$K$$ and $$(k-2)/2$$ copies of the 2-torus.

Particular cases
We use $$P$$ for the real projective plane, $$K$$ for the Klein bottle, and $$T$$ for the 2-torus.