On topological fundamental groups of quotient spaces

TL;DR

This paper investigates the topological properties of fundamental groups of quotient spaces, focusing on the density of the induced homomorphism and conditions for the quotient fundamental group to be indiscrete.

Contribution

It provides new conditions under which the topological fundamental group of a quotient space is dense or indiscrete, enhancing understanding of their topological structure.

Findings

Conditions for density of the induced homomorphism $p_*$

Criteria for $ ext{pi}_1^{qtop}(X/A,*)$ to be indiscrete

Applications to properties of quotient space fundamental groups

Abstract

Let $p:X\rightarrow X/A$ be a quotient map, where $A$ is a subspace of $X$. We explore conditions under which $p_*(\pi_1^{qtop}(X,x_0))$ is dense in $\pi_1^{qtop}(X/A,*))$, where the fundamental groups enjoy the natural quotient topology inherited from the loop space and $p_*$ is the induced continuous homomorphism by the quotient map $p$. Also, we give some applications to find out some properties for $\pi_1^{qtop}(X/A,*)$. In particular, we give some conditions in which $\pi_1^{qtop}(X/A,*)$ is an indiscrete topological group.