In my quest to understand spin representations, I am looking at the equivalent views of spin structures (on some oriented Riemannian $n$-manifold). Given such a manifold $M$, its tangent bundle $TM$ is described by gluing cocycles $g_{ab}:U_{ab}\rightarrow SO(n)$.

A *spin structure* would be some lift of $g_{ab}$ to $U_{ab}\rightarrow Spin(n)$, or in other words, that there is a $Spin(n)$-equivariant lift of the frame bundle $F_{SO}(M)\rightarrow M$ with respect to the double cover $Spin(n)\rightarrow SO(n)$.

A *spin-c structure* would likewise be a combination of a principal $U(1)$-bundle $F_{U(1)}$ and a $Spin^c(n)$-bundle $F_{spinc}$ (over $M$) with a bundle map $F_{spinc}\rightarrow F_{SO}(M)\times F_{U(1)}$.

Now apparently, assuming our manifold admits some sort of cell decomposition, a spin structure corresponds to a [homotopy class of a] trivialization of $TM\oplus \mathbb{R}^k$ over the 1-skeleton of $M$ which extends over the 2-skeleton. Here, $k$ is taken to be $0$ for $n\ge 3$, but for $n=1,2$ we stabilize the tangent bundle appropriately.

**Is there an analogous description for a spin-c structure?**