Pontryagin classes of vector bundles over $B\textrm {Sp}(n)$

Abstract:

Let $X$ be a finite skeleton of the classifying space of $\operatorname {BSp} (n),{\gamma _0} \to \operatorname {BSp} (n)$, the classifying bundle for $\operatorname {Sp} (n)$ vector bundles and $\gamma \to X$ the restriction of ${\gamma _0}$ over $X$. If $\xi \to X$ is another $\operatorname {Sp} (n)$ vector bundle, the Pontryagin classes ${p_q}(\xi )$ must be congruent to $d_1^q{p_q}(\gamma )$ modulo certain odd primes. Equality obtains if $\xi$ is the restriction over $X$ of a ${\xi _0} \to {\operatorname {BSp}}(n)$. In particular, ${\operatorname {Sp}}(m)$ vector bundles $\theta$ over ${\operatorname {BSp}}(n)$ have $p(\theta ) = 1$ if $m < n$.