Coincidence and the colouring of maps

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 phi: S-n --> Y without the antipodal coincidence property, that is, phi(x) not equal phi(-x) for all x is an element of S-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 phi: X --> Y such that phi(x) not equal phi(f(x))for all x is an element of X. Various results along these lines are obtained.