Free products of topological groups and $M$-equivalence

Abstract

In the paper we apply free products of topological groups for investigating the $M$-equivalence of Tychonoff spaces which are infinite disjoint sums of its subspaces. The main result is the following:
Let $X=\overset{\infty}{\underset{i=1}{\oplus}}X_{i}$ and for every $i\in \mathbb N$ there exists a topological group $G_{i}$ such that $F(X_{i+1})$ is topologically isomorphic to the free product $F(X_{i})$ and $G_{i}$.
Let $\{ n_{k} \}_{k=1}^{\infty}$ be an~increasing sequence of the natural numbers and let $\tilde{X}=\overset{\infty}{\underset{k=1}{\oplus}}X_{n_{k}}$. Then $X\overset{M}{\sim }\tilde{X}$.
This theorem give us many examples of topological spaces with topologically isomorphic free topological groups.

We use obtained results for constructing M-equivalent pairs and M-equivalent bundles of such spaces.