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.
Categories: math.AT