Equivariant stable homotopy theory

We will study equivariant homotopy theory for G a finite group (although this often easily generalizes to compact Lie groups). The general idea is that if we have two G-spaces X and Y , we’d like to study homotopy classes of equivariant maps between them: [X,Y ] = Map(X,Y )/htpy, where Map(X,Y ) = {f : X → Y |f(gx) = gf(x) for all g ∈ G}. In classical homotopy theory (i.e. when G is the trivial), we study CW complexes. The first question we should ask ourselves, then, is: What should a “G CW complex” be? Classically, a CW complex is a space X equipped with characteristic maps fα : D n α → X for α ∈ Sn (an indexing set) for n = 0, 1, 2, . . ., such that: