Geodesic
Abelian Groups with a Vanishing Homology Group
Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 406-409

Voir la notice de l'article provenant de la source Cambridge University Press

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 Q n , 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. 
Schafer, James A. Abelian Groups with a Vanishing Homology Group. Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 406-409. doi: 10.4153/CJM-1969-043-1
@article{10_4153_CJM_1969_043_1,
     author = {Schafer, James A.},
     title = {Abelian {Groups} with a {Vanishing} {Homology} {Group}},
     journal = {Canadian journal of mathematics},
     pages = {406--409},
     year = {1969},
     volume = {21},
     number = {1},
     doi = {10.4153/CJM-1969-043-1},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1969-043-1/}
}
TY  - JOUR
AU  - Schafer, James A.
TI  - Abelian Groups with a Vanishing Homology Group
JO  - Canadian journal of mathematics
PY  - 1969
SP  - 406
EP  - 409
VL  - 21
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1969-043-1/
DO  - 10.4153/CJM-1969-043-1
ID  - 10_4153_CJM_1969_043_1
ER  - 
%0 Journal Article
%A Schafer, James A.
%T Abelian Groups with a Vanishing Homology Group
%J Canadian journal of mathematics
%D 1969
%P 406-409
%V 21
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1969-043-1/
%R 10.4153/CJM-1969-043-1
%F 10_4153_CJM_1969_043_1

[1] 1. Cartan, H. and Eilenberg, S., Homologuai algebra (Princeton Univ. Press, Princeton, N.J., 1956). Google Scholar

[2] 2. Kurosh, A. G., Zur Zerlegung unendlicher Gruppen, Math. Ann. 106 (1932) 107–113. Google Scholar

[3] 3. MacLane, S., Homology (Academic Press, New York, 1963). Google Scholar

[4] 4. Schafer, J., On the homology ring of an abelian group, thesis, University of Chicago, 1965. Google Scholar

Cité par Sources :