Identities for Whitehead products and infinite sums

TL;DR

This paper establishes identities for Whitehead products and infinite sums in higher homotopy groups, and computes the infinite-sum closure for certain wedge spaces, providing insights into complex homotopy groups of spaces like the infinite earring.

Contribution

It derives general identities for Whitehead products and infinite sums, and computes the infinite-sum closure in specific wedge spaces, advancing understanding of higher homotopy groups.

Findings

Identities for Whitehead products and infinite sums are established.

The infinite-sum closure of Whitehead products in certain wedge spaces is explicitly computed.

A conjecture about the homotopy groups of the infinite earring space is proposed.

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 $\pi_{2n-1}(\mathbb{E}_n)$, $n\geq 2$.