Minimal homeomorphisms and topological $K$-theory

TL;DR

This paper constructs minimal homeomorphisms on complex spaces with specific K-theory or cohomology, bypassing classical obstructions, and explores their properties and implications for C*-algebras.

Contribution

It demonstrates the existence of minimal homeomorphisms on spaces with prescribed K-theory or cohomology, extending beyond classical fixed point obstructions.

Findings

Constructed minimal homeomorphisms with prescribed K-theory or cohomology.

Showed obstructions like Lefschetz fixed point theorem do not apply to general spaces.

Discussed potential applications to C*-algebras.

Abstract

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.