# On special subgroups of fundamental group

## TL;DR

This paper investigates special subgroups of the fundamental group generated by loops with certain properties, comparing them to understand their role in classifying arc connected topological spaces.

## Contribution

It introduces and analyzes subgroups of the fundamental group based on homotopy classes of loops, revealing their relationships and distinctions in topological space classification.

## Key findings

- Isomorphic subgroups imply isomorphic fundamental groups for certain spaces.
- Existence of spaces with same fundamental group but different special subgroups.
- Hierarchy of subgroups: ${rak P}^ extomega(X) extless ${rak P}^c(X) extless ext{pi}_1(X)$.

## Abstract

Suppose $\alpha$ is a nonzero cardinal number, $\mathcal I$ is an ideal on arc connected topological space $X$, and ${\mathfrak P}_{\mathcal I}^\alpha(X)$ is the subgroup of $\pi_1(X)$ (the first fundamental group of $X$) generated by homotopy classes of $\alpha\frac{\mathcal I}{}$loops. The main aim of this text is to study ${\mathfrak P}_{\mathcal I}^\alpha(X)$s and compare them. Most interest is in $\alpha\in\{\omega,c\}$ and $\mathcal I\in\{\mathcal P_{fin}(X),\{\varnothing\}\}$, where $\mathcal P_{fin}(X)$ denotes the collection of all finite subsets of $X$. We denote ${\mathfrak P}_{\{\varnothing\}}^\alpha(X)$ with ${\mathfrak P}^\alpha(X)$. We prove the following statements: $\bullet$ for arc connected topological spaces $X$ and $Y$ if ${\mathfrak P}^\alpha(X)$ is isomorphic to ${\mathfrak P}^\alpha(Y)$ for all infinite cardinal number $\alpha$, then $\pi_1(X)$ is isomorphic to $\pi_1(Y)$; $\bullet$ there are arc connected topological spaces $X$ and $Y$ such that $\pi_1(X)$ is isomorphic to $\pi_1(Y)$ but ${\mathfrak P}^\omega(X)$ is not isomorphic to ${\mathfrak P}^\omega(Y)$; $\bullet$ for arc connected topological space $X$ we have ${\mathfrak P}^\omega(X)\subseteq{\mathfrak P}^c(X) \subseteq\pi_1(X)$; $\bullet$ for Hawaiian earring $\mathcal X$, the sets ${\mathfrak P}^\omega({\mathcal X})$, ${\mathfrak P}^c({\mathcal X})$, and $\pi_1({\mathcal X})$ are pairwise distinct. So ${\mathfrak P}^\alpha(X)$s and ${\mathfrak P}_{\mathcal I}^\alpha(X)$s will help us to classify the class of all arc connected topological spaces with isomorphic fundamental groups.