In [8, 6] it was shown that for each k and n such that 2k>n, there exists a contractible k-dimensional complex Y and a continuous map ϕ[ratio ][]n→Y without the antipodal coincidence property, that is, ϕ(x)≠ϕ(−x) for all x∈[]n. In this paper it is shown that for each k and n such that 2k>n, and for each fixed-point free homeomorphism f of an n-dimensional paracompact Hausdorff space X onto itself, there is a contractible k-dimensional complex Y and a continuous map ϕ[ratio ]X→Y such that ϕ(x)≠ϕ(f(x)) for all x∈X. Various results along these lines are obtained.