In this paper, we prove parametrized Borsuk-Ulam theorems for bundles whose fibre has the same cohomology (mod p ) as a product of spheres with any free ‫ޚ‬ paction and for bundles whose fibre has rational cohomology ring isomorphic to the rational cohomology ring of a product of spheres with any free S 1 -action. These theorems extend the result proved by Koikara and Mukerjee in [7]. Further, in the particular case where G D ‫ޚ‬ p , we estimate the "size" of the ‫ޚ‬ p -coincidence set of a fibre-preserving map.

Let G D Z p k be a cyclic group of prime power order and let V and W be orthogonal representations of G withWe give an estimate for the dimension of the set f 1 f0g in terms of V and W . This extends the Bourgin-Yang version of the Borsuk-Ulam theorem to this class of groups. Using this estimate, we also estimate the size of the G -coincidences set of a continuous map from S.V / into a real vector space W 0 .

We estimate the number of homotopy types of path-components of the mapping spaces M(S m , FP n) from the m-sphere S m to the projective space FP n for F being the real numbers R, the complex numbers C, or the skew algebra H of quaternions. Then, the homotopy types of path-components of the mapping spaces M(E m , FP n) for the suspension E m of a homology m-sphere m are studied as well.

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Abstract. Let V and W be orthogonal representations of G with V G = W G = {0}. Let S(V ) be the sphere of V and f : S(V ) → W be a G-equivariant mapping. We give an estimate for the dimension of the set Z f = f −1 {0} in terms of dim V and dim W , if G is the torus T k , or the p-torus Z k p . This extends the classical Bourgin-Yang theorem onto this class of groups. Finally, we show that for any p-toral group G and a G-map f : S(V ) → W , with dim V = ∞ and dim W < ∞, we have dim Z f = ∞.