A Continuous Version of the Borsuk-Ulam Theorem

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