Let $N^3$ be a smooth manifold, and let $u:\Sigma\to N$, $v:\Sigma'\to N$ be two smooth immersions of closed orientable surfaces. (Note that $\Sigma$ and $\Sigma'$ may be of different topology). Suppose also that $u$ and $v$ are homologous. (To pass to homology, I can triangulate $\Sigma$ and $\Sigma'$ so that $u$ and $v$ now become formal sums of maps from the two-cube $[0,1]^2$ into $N^3$; moreover, these maps are not just continuous but smooth.)
Then, in homology $H_2(N,\mathbb{Z})$, we of course have that $[u]-[v]=0$, and so homology tells us that there is some $3$-cycle $\Omega$ so that in homology, $\partial\Omega = [u]-[v]$.
Question: May I insist that the three-cycle $\Omega$ be not just continuous, but in fact $C^2$, or maybe even $C^\infty$? Perhaps I need $N$ to be closed, or orientable?
That is: We know that $\Omega$ is a sum of continuous maps $e_i:[0,1]^3\to N$ from the 3-cube, $$ \Omega = \sum_i \alpha_i e_i, $$ where of course $\alpha_i\in\mathbb{Z}$. Can I insist on having greater regularity on the $e_i$?