  Using equivariant obstruction theory we construct equivariant maps from
certain classifying spaces to representation spheres for cyclic groups, product
of elementary Abelian groups and dihedral groups. Restricting them to finite
skeleta constructs equivariant maps between spaces which are related to the
topological Tverberg conjecture. This answers negatively a question of
\"Ozaydin posed in relation to weaker versions of the same conjecture. Further,
it also has consequences for Borsuk-Ulam properties of representations of
cyclic and dihedral groups.
