0
$\begingroup$

Let $\pi : E\to B$ be a fiber bundle with fiber $F$ over a finite complex $B$ whose structure group is a compact Lie group $G$. How can we determine the principal $G$-bundle associated to $\pi$? For example if we consider the bundle $S({\gamma}_{1}\times {\gamma}_{1})\to {BO}_{1}\times {BO}_{1}$ where ${\gamma}_{1}\to {BO}_{1}$ is the $1$-dimensional universal vector bundle over ${BO}_{1}$ and $S({\gamma}_{1}\times {\gamma}_{1})$ is the sphere bundle of ${\gamma}_{1}\times {\gamma}_{1}$ over ${BO}_{1}\times {BO}_{1}$, then what is the principal ${O}_{1}\times {O}_{1}$-bundle associated to $S({\gamma}_{1}\times {\gamma}_{1})\to {BO}_{1}\times {BO}_{1}$?

$\endgroup$
1
  • $\begingroup$ Concerning your last question, the principal bundle is just $EO(1)\times EO(1)$. The principal bundle comes first; you don't need to go through the vector or sphere bundles. $\endgroup$ Dec 9, 2014 at 20:14

1 Answer 1

5
$\begingroup$

The associated principal bundle is the set of pairs $(b,i)$ so that $i$ is an isomorphism of $E_b$ with $F$ as $G$-space.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.