Minimal homeomorphisms and topological $K$-theory

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. Minimal are constructed compact connected metric with any prescribed finitely generated $K$-theory or cohomology. In particular, although non-zero Euler characteristic obstructs homeomorphism CW-complex, this is case space. also allow some control map and cohomology induced from homeomorphisms. This allows construction many homotopic identity. Applications $C^\*$-algebras will be discussed in another paper.