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One way to define a spin structure on a $SO(n)$-bundle $p:E\to B$ is to require that there is an element $\sigma\in H^1(E;Z_2)$ such that restricted to each fiber gives a generator of $H^1(SO(n);Z_2)$. Then this is equivalent to say that: (1) $\sigma$ determines a double cover of $E$, such that (2) each fiber $SO(n)$ is covered by its double cover $spin(n)$.

I do not understand how the definition implies (1).

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    $\begingroup$ Real line bundles over $X$ are classified by their first Stiefel-Whitney class, which is an element of $H^1(X;\mathbb Z/2)$. Now pass to the unit sphere bundle $E$; this is a double cover $E \to X$. That's what's being referred to. (Conversely given any double cover $E \to X$ you can 'fill it in' to be a real line bundle over $X$; there is a bijection between isomorphism classes of double covers $E \to X$ and isomorphism classes of real line bundles over $X$.) $\endgroup$
    – user98602
    Jun 25 '15 at 8:03

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