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Proceedings of the American Mathematical Society

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


Authors: John Mack, Sidney A. Morris and Edward T. Ordman
Journal: Proc. Amer. Math. Soc. 40 (1973), 303-308
MSC: Primary 22A05
MathSciNet review: 0320216
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Abstract: It is shown that a free topological group on a $ {k_\omega }$-space is a $ {k_\omega }$-space. Using this it is proved that if $ X$ is a $ {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_\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.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0320216-6
Keywords: Free topological groups, $ {k_\omega }$-space, projective abelian group
Article copyright: © Copyright 1973 American Mathematical Society