Identities for Whitehead products and infinite sums

Abstract

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 $\prod_{j=1}^{\infty}\left(\pi_{n}(X_j)\otimes \prod_{k>j}\pi_n(X_k)\right)$. The insight provided by this computation motivates a conjecture about the isomorphism type of the elusive groups $\pi_{2n-1}(\mathbb{E}_n)$, $n\geq 2$.