On the discontinuity of the π1-action

We show the classical π1-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 π1(X)×πn(X)→πn(X) fails to be continuous for a one-point union X=A∨Hn where A is an aspherical space such that π1(A) is a topological group and Hn is the (n−1)-connected, n-dimensional Hawaiian earring space Hn for which πn(Hn) is a topological abelian group.