Representations and Homotopy Theory

Let X = ΣY be a suspension. A question in homotopy theory is how to decompose the n-fold self smash product X into a wedge of spaces. Consider the set of homotopy classes [X, X]. Let the symmetric group Sn act on X by permuting positions. So for each σ ∈ Sn there is a map σ : X → X. This gives a function θ : Sn → [X, X]. By assuming that X is a suspension, [X, X] is an abelian group and so there is an extension θ : Z(Sn)→ [X, X]. Furthermore, assume that X is a p-local suspension, the map θ extends to a map θ : Z(p)(Sn)→ [X, X], where Z(p) is the set of p-local integers, that is the rational numbers a/b with b 6≡ 0 mod p. (Note: [X, X] is semi-ring, that is, h ◦ (f + g) = h◦f +h◦g holds but (f +g)◦h 6= f ◦h+g ◦h in general.) The map θ is a morphism of semi-rings. Now let 1 = ∑ α eα is a decomposition of the identity into orthogonal idempotents in Z(p)(Sn). Now consider the map eα : X → X. We have eα ◦ eα ' eα : X → X. Define a space eα(X ) by the homotopy colimit