Construct a generic unorientable surfaces of genus $g$

Question

0. In terms of topology (homology, homotopy, etc), the most general orientable 2d surfaces are the genus $g$ Riemann surfaces.

1. Is it true that an unorientable surfaces of genus $g$ is the most general unorientable 2d surfaces that we can encounter? (How to prove it?)

2. How to construct such unorientable surfaces of genus $g$? I suppose $g=1$ is $RP^2$ and $g=2$ is Klein bottle. Can we have $g=0$ and all the other integer $g$?

Answer 1

Classification of surfaces: every connected, closed surface (i.e. two-dimensional manifold) is diffeomorphic to either a sphere, a connected sum of $k > 0$ tori, or a connected sum of $\ell > 0$ real projective planes.

The first two possibilities corresponds to orientable manifolds, while the last possibility corresponds to non-orientable manifolds. In particular, a connected, closed, non-orientable surface is diffeomorphic to the connected sum of $\ell$ copies of $\mathbb{RP}^2$ for some $\ell > 0$.

For connected $R$-orientable surfaces $M_1, M_2$, we have $$H_1(M_1\# M_2; R) \cong H_1(M_1; R)\oplus H_1(M_2; R)$$ by Mayer-Vietoris. So if $M_1, M_2$ are connected, closed surfaces which are either both orientable or both non-orientable, then $g(M_1\# M_2) = g(M_1) + g(M_2)$ where $g$ denotes the genus function.

Note that $g(T^2) = 1$ and $g(\mathbb{RP}^2) = 1$, so a the connected sum of $k$ tori has genus $k$, and a connected sum of $\ell$ real projective planes has genus $\ell$. In particular, there are no non-orientable surfaces of genus zero (the only surface of genus zero is $S^2$, which is orientable).