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.

13 pages

References in corpus (5)

• The fundamental group as a topological group
• The topological fundamental group and free topological groups
• Discreteness and Homogeneity of the Topological Fundamental Group
• On Topological Homotopy Groups of $n$-Hawaiian like spaces
• Compactly generated quasitopological homotopy groups with discontinuous multiplication

Cited by in corpus (2)

• A natural pseudometric on homotopy groups of metric spaces
• On zero dimensional sequential spaces