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.