Identities for Whitehead products and infinite sums

Whitehead products and natural infinite sums are prominent in the higher homotopy groups of the n-dimensional infinite earring space Eₙ and other locally complicated Peano continua. In this paper, we derive general identities for how these operations interact with each other. As an application, we consider a shrinking-wedge X of $(n-1)$-connected finite CW-complexes X₁,X₂,X₃, and compute the infinite-sum closure W₂ₙ₋₁(X) of the set of Whitehead products [α,β] in π₂ₙ₋₁(X) where α,β∈πₙ(X) are represented in respective sub-wedges that meet only at the basepoint. In particular, we show that W₂ₙ₋₁(X) is canonically isomorphic to ∏ⱼ₌₁∞(πₙ(Xⱼ)⊗ ∏k>jπₙ(Xₖ)). The insight provided by this computation motivates a conjecture about the isomorphism type of the elusive groups π₂ₙ₋₁(Eₙ), n≥ 2.