Free topological groups and dimension

Abstract:

For a completely regular space X we denote by $F(X)$ and $A(X)$ the free topological group of X and the free Abelian topological group of X, respectively, in the sense of Markov and Graev. Let X and Y be locally compact metric spaces with either $A(X)$ topologically isomorphic to $A(Y)$ or $F(X)$ topologically isomorphic to $F(Y)$. We show that in either case X and Y have the same weak inductive dimension. To prove these results we use a Fundamental Lemma which deals with the structure of the topology of $F(X)$ and $A(X)$. We give other results on the topology of $F(X)$ and $A(X)$ and on the position of X in $F(X)$ and $A(X)$.