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Extensions of locally compact abelian groups. II


Authors: Ronald O. Fulp and Phillip A. Griffith
Journal: Trans. Amer. Math. Soc. 154 (1971), 357-363
MSC: Primary 18.20; Secondary 22.00
DOI: https://doi.org/10.1090/S0002-9947-1971-0272870-8
Part I: Trans. Amer. Math. Soc. (1971), 341-356
MathSciNet review: 0272870
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Abstract: It is shown that the extension functor defined on the category $ \mathcal{L}$ of locally compact abelian groups is right-exact. Actually $ {\text{Ext}^n}$ is shown to be zero for all $ n \geqq 2$. Various applications are obtained which deal with the general problem as to when a locally compact abelian group is the direct product of a connected group and a totally disconnected group. One such result is that a locally compact abelian group G has the property that every extension of G by a connected group in $ \mathcal{L}$ splits iff $ G = {(R/Z)^\sigma } \oplus {R^n}$ for some cardinal $ \sigma $ and positive integer n.


References [Enhancements On Off] (What's this?)

  • [1] R. Baer, Erweiterung von Gruppen und ihren Isomorphismen, Math. Z. 38 (1934), 375-416. MR 1545456
  • [2] S. Eilenberg and S. Mac Lane, Group extensions and homology, Ann. of Math. (2) 43 (1942), 757-831. MR 4, 88. MR 0007108 (4:88d)
  • [3] R. Fulp and P. Griffith, Extensions of locally compact abelian groups. I, Trans. Amer. Math. Soc. 154 (1970), 341-356.
  • [4] D. K. Harrison, Infinite abelian groups and homological methods, Ann. of Math. (2) 69 (1959), 366-391. MR 21 #3481. MR 0104728 (21:3481)
  • [5] E. Hewitt and K. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations, Die Grundlehren der math. Wissenschaften, Band 115, Academic Press, New York and Springer-Verlag, Berlin, 1963. MR 28 #158. MR 551496 (81k:43001)
  • [6] G. Hochschild, Group extensions of Lie groups. I, Ann. of Math. (2) 54 (1951), 96-109. MR 13, 12. MR 0041858 (13:12b)
  • [7] -, Group extensions of Lie groups. II, Ann. of Math. (2) 54 (1951), 537-551. MR 13, 318. MR 0043789 (13:318d)
  • [8] S. Mac Lane, Homology, Die Grundlehren der math. Wissenschaften, Band 114, Academic Press, New York and Springer-Verlag, Berlin, 1963. MR 28 #122.
  • [9] M. Moskowitz, Homological algebra in locally compact abelian groups, Trans. Amer. Math. Soc. 127 (1967), 361-404. MR 35 #5861. MR 0215016 (35:5861)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0272870-8
Keywords: Locally compact abelian groups, topological group extensions, right exactness of Ext, $ {\text{Ext}^n} = 0$, split exact sequences, connected by totally disconnected, totally disconnected by connected
Article copyright: © Copyright 1971 American Mathematical Society

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