(Equivariant) Homotopy in Algebraic Topology

An important focus in algebraic topology is homotopy theory, which focuses on a particular set of invariants spaces carry, namely, groups \pi_n(X) for all n\geq 0 where X is some space. We can modify this setting in three ways: making it equivariant, stabilizing, or making it equivariant and stabilizing. We will introduce homotopy and then see how this invariant changes in the aforementioned settings.