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.

Value of $k$	Surface name	First expression as connected sum (in terms of $P,K,T$)	Alternate expressions as connected sum (in terms of $P,K,T$)
1	real projective plane	$P$	--
2	Klein bottle	$P\#P$	$K$
3	Dyck's surface	$P\#P\#P$	$P\#K$, $P\#T$
4	?	$P\#P\#P\#P$	$P\#P\#K$, $P\#P\#T$, $K\#K$, $K\#T$

Algebraic topology

Homology

Further information: homology of compact non-orientable surfaces

Cohomology

Further information: cohomology of compact non-orientable surfaces

Homotopy

Further information: homotopy of compact non-orientable surfaces