Recently I asked on stack exchange the following question: https://math.stackexchange.com/questions/2088888/vanishing-of-certain-cohomology-class-and-existence-of-spin-structure

I would like to know whether there is a similar construction for the spin$^c$ manifolds: namely whether one can construct some cohomology class indpendent from the choice of transition functions and liftings to $spin^c(n)$ with the property that this class is trivial iff $M$ is spin$^c$ manifold.

Remark: I know $C^*$-algebraic approach which gives the so called Dixmier-Douady class in $H^3(M,\mathbb{Z})$. However I would like to understand whether one can proceed purely geometrically.