  The Lefschetz fixed point theorem provides a powerful obstruction to the
existence of minimal homeomorphisms on well-behaved spaces such as finite
CW-complexes. We show that these obstructions do not hold for more general
spaces. More precisely, minimal homeomorphisms are constructed on space with
prescribed $K$-theory or cohomology. We also allow for some control of the map
on $K$-theory and cohomology induced from these minimal homeomorphisms. This
allows for the construction of many minimal homeomorphisms that are not
homotopic to the identity. Applications to $C^*$-algebras will be discussed in
another paper.
