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?**



Spin-c Structures viewed w.r.t. Cell Decomposition
A spin-c structure on a vector bundle $\pi : E \to B$ can be thought of as two things:


(1) A complex line bundle $c : C \to B$


together with


(2) A spin structure on $\pi \oplus c : E \oplus C \to B$. $\pi \oplus c$ is meant to be the fibrewise direct sum. 


This appears in the Gompf reference given in the comments above. It's also on the Wikipedia page: <http://en.wikipedia.org/wiki/Spin_structure>


So from an obstruction theory point of view, you can view a complex spin structure on a vector bundle as (1) an element of $H^2(B;\mathbb Z)$ (a cocycle) together with (2) a trivialization of the vector bundle $\pi$ over the 1-skeleton $B^1$ such that the obstruction to extending over the 2-skeleton agrees modulo two with the value of the above cocycle. 


(1) is because line bundles over $B$ are given by maps to $K(\mathbb Z, 2)$, i.e. 2-dimensional cohomology classes. 


Note, Paul Siegel's comment is explained the the Gompf reference, as well. I prefer the above formalism, myself. 


