A stroll in equivariant $K$-theory

Abstract: Equivariant $K$-theory is a generalized equivariant cohomology theory which is a hybrid of the $K$-theory of a topological space and the representation theory of the group acting on it. In this article, we review the basics of equivariant $K$-theory and focus on the localization theorem and formula due to Atiyah and Segal, which have become important tools in equivariant topology nowadays. We then discuss the application of equivariant $K$-theory to equivariant formality, and briefly mention some recent developments.