On the Spanier Groups and Covering and Semicovering Spaces

Abstract

For a connected, locally path connected space $X$, let $H$ be a subgroup of the fundamental group of $X$, ${\pi }_1(X,x)$. We show that there exists an open cover $\mathcal{U}$ of $X$ such that $H$ contains the Spanier group $π({\U},x)$ if and only if the core of $H$ in ${\pi }_1(X,x)$ is open in the quasitopological fundamental group ${\pi }_1^{qtop}(X,x)$ or equivalently it is open in the topological fundamental group ${\pi }_1^{\tau }(X,x)$. As a consequence, using the relation between the Spanier groups and covering spaces, we give a classification for connected covering spaces of $X$ based on the conjugacy classes of subgroups with open core in ${\pi }_1^{qtop}(X,x)$. Finally, we give a necessary and sufficient condition for the existence of a semicovering. Moreover, we present a condition under which every semicovering of $X$ is a covering.

11 pages,1 figure