Below is a brief introduction to spin$^{\mathbf{C}}$ structure that I took from Wikipedia. For more information, one should refer to https://en.wikipedia.org/wiki/Spin_structure#SpinC_structures.

A spin$^{\mathbf{C}}$ structure is analogous to a spin structure for orientable Riemannian manifold, but uses the spin$^{\mathbf{C}}$ group, which is defined by the exact sequence \begin{equation} 1 \rightarrow \mathbf{Z}_2 \rightarrow \text{spin}^{\mathbf{C}}(n) \rightarrow SO(n)\times U(1) \rightarrow 1. \end{equation}

A spin$^{\mathbf{C}}$ structure exists iff the third integral Stiefel-Whitney class of the manifold vanishes. Moreover, the set of spin$^{\mathbf{C}}$ structures has a free transitive action of $H^2(M, \mathbf{Z})$. Thus, spin$^{\mathbf{C}}$ correspond to elements of $H^2(M, \mathbf{Z})$ although not in a natural way.

For spin structure on an orientable two dimensional manifold equipped with a triangulation, there is a nice combinatorial representation in terms of the Kasteleyn orientation (i.e. orientations of edges so that every face has an odd number of clockwise oriented edges). This correspondence is detailed in https://arxiv.org/abs/math-ph/0608070. See also this post Combinatorial spin structures.

I would like to know if there is a similar combinatorial representation of spin$^{\mathbf{C}}$ structures on orientable manifolds in two dimension.

naturallyisomorphic to $\mathbb{Z}$. $\endgroup$ – Alex Degtyarev Nov 7 '17 at 19:26