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 ${\mathbb{E}}_n$ 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 finite $(n-1)$-connected CW-complexes $X_1,X_2,X_3,\dots$ and compute the infinite-sum closure ${\mathcal{W}}_{2n-1}(X)$ of the set of Whitehead products $[\alpha ,\beta ]$ in ${\pi }_{2n-1}\left(X\right)$ where $\alpha ,\beta \in {\pi }_n(X)$ are represented in respective sub-wedges that meet only at the basepoint. In particular, we show that ${\mathcal{W}}_{2n-1}(X)$ is canonically isomorphic to ${\prod }_{j=1}^{\mathrm{\infty }}({\pi }_n(X_j)\otimes {\prod }_{k>j}{\pi }_n(X_k))$. The insight provided by this computation motivates a conjecture about the isomorphism type of the elusive groups ${\pi }_{2n-1}({\mathbb{E}}_n)$, $n\ge 2$.