On the discontinuity of the $\pi_1$-action

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.