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 $E_n$ 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

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