Given an element in the (first) homology group of a surface, I would like to know if it can be represented as a simple closed curve. For orientable surfaces, this is well-known, but I wasn't able to find a reference for non-orientable surfaces.
For orientable surfaces, the sphere with $g$ handles has homology $\mathbb{Z}^{2g}$, and an element $(a_1, \dots, a_{2g}) \in \mathbb{Z}^{2g}$ can be represented by a simple closed curve if and only if $gcd(a_1, \dots, a_{2g})=1$. This is classical and actually not too hard to prove.
For non-orientable surfaces, the sphere with $k$ crosscaps has homology $\mathbb{Z}_2 \times \mathbb{Z}^{k-1}$. Let $(a, b_1, \dots, b_{k-1}) \in \mathbb{Z}_2 \times \mathbb{Z}^{k-1}$. If $k$ is odd and $gcd(b_1, \dots, b_{k-1})=1$, then we can represent this element as a simple curve, regardless of the value of $a$ by the orientable case. But there are other homology classes not of this form which can be represented by simple curves. For example, I think $(0,2,0) \in \mathbb{Z}_2 \times \mathbb{Z}^2$ can be represented by a simple curve on the sphere with 3 crosscaps. Incidentally, while I am at it, I'd also like to know why it is common to use $\mathbb{Z}_2$-homology when working with non-orientable surfaces, as opposed to $\mathbb{Z}$-homology (which is what I am using).
