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Abelian Groups with a Vanishing Homology Group

Published online by Cambridge University Press:  20 November 2018

James A. Schafer
James A. Schafer*
Affiliation:
University of Michigan, Ann Arbor, Michigan
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In this paper, we wish to characterize those abelian groups whose integral homology groups vanish in some positive dimension. We obtain a complete characterization provided the dimension in which the homology vanishes is odd; in fact, we prove that the only abelian groups which possess a vanishing homology group in an odd dimension are, up to isomorphism, subgroups of Qn, where Q denotes the additive group of rational numbers. The case of vanishing in an even dimension is much more complicated. We exhibit a class of groups whose homology vanishes in even dimensions and is otherwise very nice, namely the subgroups of Q/Z, and then show that unless we impose further restrictions, there exist abelian groups which possess the homology of subgroups of Q/Z without being isomorphic to a subgroup of Q/Z.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 406 - 409
Copyright
Copyright © Canadian Mathematical Society 1969

References

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3. MacLane, S., Homology (Academic Press, New York, 1963).Google Scholar
4. Schafer, J., On the homology ring of an abelian group, thesis, University of Chicago, 1965.Google Scholar