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).
Best Answer
The complete answer follows from the result of the paper referenced below (as the math review points out the result was also obtained slightly earlier by McCarthy and Pinkall).
@article {MR2161731, AUTHOR = {Gadgil, Siddhartha and Pancholi, Dishant}, TITLE = {Homeomorphisms and the homology of non-orientable surfaces}, JOURNAL = {Proc. Indian Acad. Sci. Math. Sci.}, FJOURNAL = {Indian Academy of Sciences. Proceedings. Mathematical Sciences}, VOLUME = {115}, YEAR = {2005}, NUMBER = {3}, PAGES = {251--257}, ISSN = {0253-4142}, MRCLASS = {57M60 (20F38 57N05)}, MRNUMBER = {2161731 (2006f:57019)}, MRREVIEWER = {Mustafa Korkmaz}, DOI = {10.1007/BF02829656}, URL = {http://dx.doi.org/10.1007/BF02829656}, }