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reduction of structure group

• For $\xi = (E \xrightarrow{p} B)$ a real $n{\hbox{-}}$plane bundle, $E$ is orientable iff $?$
• A principal ${\operatorname{SO}}_n$ bundle lifts to a principal ${\operatorname{Spin}}_n$ bundle iff the 2nd Stiefel-Whitney class vanishes, $w_2(P) \in H^2(X; { \mathbf{F} }_2)$.
  • Choices of spin structures biject with $H^1(X; { \mathbf{F} }_2)$.