spin structure definition

Question

Suppose we have a principal $SO(n)$-bundle $E$ over $B$, with projection map $p$. We say that it admits a spin structure if there is a prinicipal $spin(n)$-bundle $E'$ over B, with projection map $p'$, and a map $\phi:E'\to E$, such that it is the double cover restricted to each fiber, and $p'=p o \phi$. Assuming this definition is correct, I am thinking that we can always find a spin structure: let $E'$ be a bundle over $B$ with the same transition maps as for $E$ and the same way they act on each element $g$ of $SO(n)$, the transition functions can act on the two inverse images of $g$ in $spin(n)$. Is this not correct?

Answer 1

The problem is, "the same way" isn't well-defined.

To specify a transition map for an $SO(n)$ bundle is to specify an element of $SO(n)$ to multiply by at each fiber in the intersection. In order to lift it to a $Spin(n)$ bundle, you have to pick which of the two preimages of that element you want (at one point, and so everywhere in that intersection by continuity). This is locally always possible, but you may find it impossible to reconcile your choices globally.