Cohomology theories for highly structured ring spectra

Abstract

This is a survey paper on cohomology theories for $A_{\infty }$ and $E_{\infty }$ ring spectra. Different constructions and main properties of topological André-Quillen cohomology and of topological derivations are described. We give sample calculations of these cohomology theories and outline applications to the existence of $A_{\infty }$ and $E_{\infty }$ structures on various spectra. We also explain the relationship between topological derivations, spaces of multiplicative maps and moduli spaces of multiplicative structures.