Let $ E$ be a spin$ ^c$ bundle and $ spin^c(E)$ the corresponding $ spin^c(n)$ -principial bundle. Let $ g_{U,V}: U \cap V \to spin^c(n)$ denote transition functions for this principial bundle and consider the map $ \nu:spin^c(n) \to \mathbb{T}$ defined by $ \nu(w)=w^!w$ where $ (v_1 \cdot … \cdot v_r)^!=v_r \cdot … \cdot v_1$ . We can form the composition $ \nu \circ g_{U,V}:U \cap V \to \mathbb{T}$ . As this satisfies cocycle property we can form the line bundle $ L_E$ .
Let us assume that $ E$ is $ spin^c$ but is not spin.
How to prove that the first Chern class of $ L_E$ is odd (in the sense that $ j^*(c_1(L_E)) \neq 0 $ where $ j^*$ is mod 2 reduction of coefficients)?
