A continuous version of the Borsuk-Ulam theorem

Abstract:

Let $p:E \to B$ be an $n$-sphere bundle, $q:V \to B$ be an ${{\mathbf {R}}^n}$-bundle and $f:E \to V$ be a fibre preserving map over a paracompact space $B$. Let $\overline p :\overline E \to B$ be the projectivized bundle obtained from $p$ by the antipodal identification and let ${\overline A _f}$ be the subset of $\overline E$ consisting of pairs $\{ e, - e\}$ such that $fe = f( - e)$. If the cohomology dimension $d$ of $B$ is finite then the map $(\bar {p} | \overline {A}_f)^*$ is injective for a continuous cohomology theory ${H^*}$. Moreover, if the $j$th Stiefel-Whitney class of $q$ is zero for $1 \leqslant j \leqslant r$ then $(\bar {p} | \overline {A}_f)^*$ is injective in degrees $i \geqslant d - r$. If all the Stiefel-Whitney classes of $q$ are zero then $(\bar {p} | \overline {A}_f)^*$ is injective in every degree.