Abstract.It is shown that a free topological group on a kaspace is a ArM-space. Using this it is proved that if A" is a Ar^-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 Ä^-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 ¡s exactly one. Introduction.In [2] Graev showed that a free topological group on a compact Hausdorff space is a Acra-space. It was observed [6] that Graev's proof worked not only for free topological groups but also for groups with the maximum topology relative to a given compact set, and several applications to free products of topological groups were obtained [8],

Abstract.It is shown that a free topological group on a kaspace is a ArM-space. Using this it is proved that if A" is a Ar^-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 Ä^-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 ¡s exactly one. Introduction.In [2] Graev showed that a free topological group on a compact Hausdorff space is a Acra-space. It was observed [6] that Graev's proof worked not only for free topological groups but also for groups with the maximum topology relative to a given compact set, and several applications to free products of topological groups were obtained [8],

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