Universal minimal flows from a homotopical perspective

Abstract We study universal minimal flows of connected and locally contractible topological groups by means of tools from homotopy theory and dynamical cohomology. We provide methods for computing their first cohomotopy groups and illustrate them by computations related to connected Lie groups and identity components of homeomorphism groups of compact connected surfaces with or without boundary. Since metrizable spaces have countable first cohomotopy groups, these methods serve as a useful new instrument for proving nonmetrizability of various universal minimal flows. In this way we reprove several results in this direction known before and also prove some new ones. Finally, we indicate how our methods can be used to construct finite-dimensional minimal spaces for infinite-dimensional topological groups, having all orbits of first category.