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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?

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I have been told that a $spin^c$ structure is the same thing as a choice of (a homotopy class of) a complex structure on the 2-skeleton which extends to the 3-skeleton, but I don't know how to prove this and I can't seem to find a reference. –  Paul Siegel Feb 7 '12 at 2:31
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R. Gompf. Spin^c structures and homotopy equivalences. GEometry and Toplogy Vol 1 (1997) 41--50. Is a reference for what you're looking for, Chris. –  Ryan Budney Feb 7 '12 at 2:38
    
Cooool, Thanks! –  Chris Gerig Feb 8 '12 at 5:08
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1 Answer

up vote 4 down vote accepted

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.

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