Let X$X$ be a simply connected finite CW complex, \xi$\xi$ and \eta$\eta$ vector bundles over X$X$ of the same dimensions and their dimension is big enough, so they are stable bundles. Let p$p$ be a prime. Are the following two conditions equivalent?
J(\xi)$J(\xi)$ and J(\eta)$J(\eta)$ as elements in J(X)$J(X)$ have equal p$p$-primary components. (That is there is a q$q$ not divisible by p$p$ such that q(J(\xi) - J(\eta) ) = 0.$q(J(\xi) - J(\eta) ) = 0.$)
The sphere bundles S(\xi)$S(\xi)$ and S(\eta)$S(\eta)$ are fiberwise p$p$-equivalent. (That is there is a fiberwise map S(\xi) \to S(\eta)$S(\xi) \to S(\eta)$ that induces isomorphism in homologies with Z/pZ$Z/pZ$ coefficients.)