On the discontinuity of the $\pi_1$-action

TL;DR

This paper demonstrates that the classical action of the fundamental group on higher homotopy groups can be discontinuous when these groups are given their natural quotient topology, especially in certain union spaces.

Contribution

It provides the first examples showing the $ abla_1$-action on homotopy groups can be discontinuous in the quotient topology setting.

Findings

The $ abla_1$-action fails to be continuous in specific union spaces.

The action is discontinuous for spaces involving Hawaiian earrings and aspherical components.

The paper identifies conditions under which the classical $ abla_1$-action is not topologically continuous.

Abstract

We show the classical $\pi_1$-action on the $n$-th homotopy group can fail to be continuous for any $n$ when the homotopy groups are equipped with the natural quotient topology. In particular, we prove the action $\pi_1(X)\times\pi_n(X)\to\pi_n(X)$ fails to be continuous for a one-point union $X=A\vee \mathbb{H}_n$ where $A$ is an aspherical space such that $\pi_1(A)$ is a topological group and $\mathbb{H}_n$ is the $(n-1)$-connected, n-dimensional Hawaiian earring space $\mathbb{H}_n$ for which $\pi_n(\mathbb{H}_n)$ is a topological abelian group.