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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 \(H^1(X; { \mathbf{F} }_2)\).