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.

29 pages