A core-free semicovering of the Hawaiian Earring

TL;DR

This paper constructs a semicovering of the Hawaiian Earring with an open subgroup of its fundamental group that contains no nontrivial normal subgroup, challenging existing classification frameworks.

Contribution

It provides the first example of a core-free semicovering of the Hawaiian Earring, expanding understanding of semicovering space classifications.

Findings

Existence of a semicovering with a core-free open subgroup.

Counterexample to previous classification assumptions.

Highlights complexity of fundamental group structures.

Abstract

The connected covering spaces of a connected and locally path-connected topological space $X$ can be classified by the conjugacy classes of those subgroups of $\pi_1(X,x)$ which contain an open normal subgroup of $\pi_1(X,x)$, when endowed with the natural quotient topology of the compact-open topology on based loops. There are known examples of semicoverings (in the sense of Brazas) that correspond to open subgroups which do not contain an open normal subgroup. We present an example of a semicovering of the Hawaiian Earring $\mathds{H}$ with corresponding open subgroup of $\pi_1(\mathds{H})$ which does not contain {\em any} nontrivial normal subgroup of $\pi_1(\mathds{H})$.