On special subgroups of fundamental group
Fatemah Ayatollah Zadeh Shirazi, Fatemeh Ebrahimifar, Mohammad Ali Mahmoodi
Abstract
Suppose $α$ is a nonzero cardinal number, $\mathcal I$ is an ideal on arc connected topological space $X$, and ${\mathfrak P}_{\mathcal I}^α(X)$ is the subgroup of $π_1(X)$ (the first fundamental group of $X$) generated by homotopy classes of $α\frac{\mathcal I}{}$loops. The main aim of this text is to study ${\mathfrak P}_{\mathcal I}^α(X)$s and compare them. Most interest is in $α\in\{ω,c\}$ 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 ${\mathfrak P}_{\{\varnothing\}}^α(X)$ with ${\mathfrak P}^α(X)$. We prove the following statements: $\bullet$ for arc connected topological spaces $X$ and $Y$ if ${\mathfrak P}^α(X)$ is isomorphic to ${\mathfrak P}^α(Y)$ for all infinite cardinal number $α$, then $π_1(X)$ is isomorphic to $π_1(Y)$; $\bullet$ there are arc connected topological spaces $X$ and $Y$ such that $π_1(X)$ is isomorphic to $π_1(Y)$ but ${\mathfrak P}^ω(X)$ is not isomorphic to ${\mathfrak P}^ω(Y)$; $\bullet$ for arc connected topological space $X$ we have ${\mathfrak P}^ω(X)\subseteq{\mathfrak P}^c(X) \subseteqπ_1(X)$; $\bullet$ for Hawaiian earring $\mathcal X$, the sets ${\mathfrak P}^ω({\mathcal X})$, ${\mathfrak P}^c({\mathcal X})$, and $π_1({\mathcal X})$ are pairwise distinct. So ${\mathfrak P}^α(X)$s and ${\mathfrak P}_{\mathcal I}^α(X)$s will help us to classify the class of all arc connected topological spaces with isomorphic fundamental groups.