Free topological groups and the projective dimension of a locally compact abelian group

Abstract

It is shown that a free topological group on a k ω {k_\omega } -space is a k ω {k_\omega } -space. Using this it is proved that if X X is a k ω {k_\omega } -group then it is a quotient of a free topological group by a free topological group. A corollary to this is that the projective dimension of any k ω {k_\omega } -group, relative to the class of all continuous epimorphisms admitting sections, is either zero or one. In particular the projective dimension of a connected locally compact abelian group or a compact abelian group is exactly one.