On Subgroup Topologies on Fundamental Groups

Abstract

It is important to classify covering subgroups of the fundamental group of a topological space using their topological properties in the topologized fundamental group. In this paper, we introduce and study some topologies on the fundamental group and use them to classify coverings, semicoverings, and generalized coverings of a topological space. To do this, we use the concept of subgroup topology on a group and discuss their properties. In particular, we explore which of these topologies make the fundamental group a topological group. Moreover, we provide some examples of topological spaces to compare topologies of fundamental groups.

On Generalized Covering Groups of Topological Groups

“…α 1 ∼ H α 2 if and only if α 1 (1) = α 2 (1) and [α 1 * α 2 −1 ] ∈ H. The equivalence class of α is denoted by α H . One can consider the quotient space X H = P (X, x 0 )/ ∼ H and the endpoint projection map p H : ( X H , e H ) → (X, x 0 ) defined by α H → α (1), where e H is the class of the constant path at x 0 .…”

“…If α ∈ P (X, x 0 ) and U is an open neighbourhood of α(1), then a continuation of α in U is a path β = α * γ, where γ is a path in U with γ(0) = α (1). Put N( α H , U) = { β H ∈ X H | β is a continuation of α in U}.…”