From $0\rightarrow \mathbb{Z}\rightarrow \mathbb{Z}\rightarrow \mathbb{Z}\_{2}\rightarrow 0$, we have the long exact sequence 
$H^{1}(X,\mathbb{Z})\rightarrow H^{1}(X,\mathbb{Z}\_{2})\rightarrow H^{2}(X,\mathbb{Z})$. Meanwhile 
$H^{1}(X,\mathbb{Z}\_{2})\cong H^{1}(X,\mathbb{Z})\otimes\mathbb{Z}\_{2} \oplus Tor(H^{2}(X,\mathbb{Z}),\mathbb{Z}\_{2})$.


My question is: if $L$ is a real line bundle on $X$, does $w\_{1}(L)$ have component in $H^{1}(X,\mathbb{Z})\otimes\mathbb{Z}\_{2}$ or $Tor(H^{2}(X,\mathbb{Z}),\mathbb{Z}\_{2})$ under the above isomorphism.



One question on first Stiefel-Whitney class 
Every element in $H^1(X,\mathbb Z/2)$ is the first Stiefel-Whitney class of exactly one line bundle. This is just because $H^1(X,\mathbb Z/2)$ parameterizes double covers of $X$, and for every double cover there is a unique line bundle that trivializes on it. So $w\_1(L)$ can have components in either, neither, or both.


