On special subgroups of fundamental group

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.