Group Actions and Chromatic Homotopy Theory

Abstract We give a survey on the theory of group actions on finite complexes, with a focus on its homological aspects. We describe the passage from classical Smith theory on the mod $p$ p homology of fixed point spaces for periodic maps to recent refinements in terms of generalized homology theories, in particular the Morava K-theories from chromatic homotopy theory. We also discuss the relationship between these results and the structure of the moduli stack of equivariant formal groups via complex bordism theory.