In an attempt to extend the parametrized version of the Borsuk-Utam theorem as obtained by Dold [2] using characteristic polynomials in the situation expressed in the title of this paper we tried to use the Bredon operation and Conner-Miller classes, as was used by Kahn [3] and Lin [5], [6] in proving a Borsuk-Ulam theorem for maps of manifolds. Consequently we have obtained a Borsuk-Ulam theorem for maps of fibre bundles with manifolds as fibres, and for particular manifolds like product of spheres this procedure gives a better estimate of the Borsuk-Ulam set of the map of the bundle than that obtained using the method of characteristic polynomials (cf. [4]).In Section 1 we state our main theorem and consequences. In Section 2 we prove the theorem.We would like to thank the referee for suggesting this shortened and improved presentation of the paper, and also for making several useful suggestions.
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