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Free Abelian topological groups and the Pontryagin-Van Kampen duality

Published online by Cambridge University Press:  17 April 2009

Vladimir Pestov
Vladimir Pestov
Affiliation:
Department of Mathematics, Victoria University of Wellington, Wellington, New Zealand, e-mail: [email protected]
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Abstract

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We study the class of Tychonoff topological spaces such that the free Abelian topological group A(X) is reflexive (satisfies the Pontryagin-van Kampen duality). Every such X must be totally path-disconnected and (if it is pseudocompact) must have a trivial first cohomotopy group π1(X). If X is a strongly zero-dimensional space which is either metrisable or compact, then A(X) is reflexive.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 52 , Issue 2 , October 1995 , pp. 297 - 311
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Arhangel'skii˘, A.V., ‘Relations among invariants of topological groups and their subspaces’, Russian Math. Surveys 35 (1980), 1–23.CrossRefGoogle Scholar
[2]Arhangel'skii˘, A.V., ‘Linear homeomorphisms of function spaces’, Soviet Math. Dokl. 25 (1982), 852–855.Google Scholar
[3]Banaczcyk, W., Additive subgroups of topological vector spaces, Lecture Notes in Mathematics 1466 (Springer-Verlag, Berlin, Heidelberg, New York, 1991).CrossRefGoogle Scholar
[4]Blasco, J.L., ‘On μ-spaces and kR-spaces’, Proc. Amer. Math. Soc. 67 (1977), 179–186.Google Scholar
[5]Engelking, R., General topology (PWN, Warczawa, 1977).Google Scholar
[6]Flood, J., Free topological vector spaces, (Ph.D. thesis) (Australian National University, Canberra, 1975).Google Scholar
[7]Flood, J., Free locally convex spaces, Dissert. Math. CCXXI (PWN, Warsczawa, 1984).Google Scholar
[8]Fuks, D.B. and Rokhlin, V.A., Beginner's Course in Topology: Geometric Chapters, (translated from the Russian by Iacob, A.) (Springer-Verlag, Berlin, Heidelberg, New York, 1984).CrossRefGoogle Scholar
[9]Gelbaum, B.R., ‘Free topological groups’, Proc. Amer. Math. Soc. 12 (1961), 737–743.CrossRefGoogle Scholar
[10]Hewitt, E. and Ross, K.A., Abstract harmonic analysis. I, (second edition) (Springer-Verlag, Berlin, Heidelberg, New York, 1979).CrossRefGoogle Scholar
[11]Hu, S.-T., Homotopy theory, Pure and Applied Mathematics, 8 (Academic Press, New York, London, 1959).Google Scholar
[12]Kaplan, S., ‘Extensions of the Pontryagin duality, I. Infinite products’, Duke Math. J. 15 (1948), 649–658.CrossRefGoogle Scholar
[13]Kirillov, A.A. and Gvishiani, A.D., Theorems and problems in functional analysis, (translated from Russian by McFaden, Harold H.) (Springer-Verlag, Berlin, Heidelberg, New York, 1982).CrossRefGoogle Scholar
[14]Markov, A.A., ‘Three papers on topological groups’, Amer. Math. Soc. Transl. 30 (1950), 120 pp.Google Scholar
[15]Martin-Peinador, E., ‘A reflexive admissible topological group must be locally compact’, Proc. Amer. Math. Soc. (to appear).Google Scholar
[16]Morris, S.A., Pontryagin duality and the structure of locally compact Abelian groups (Cambridge University Press, Cambridge, London, New York, Melbourne, 1977).CrossRefGoogle Scholar
[17]Morris, S.A., ‘Free Abelian topological groups’, in Categorical Topology, Proc. Conference Toledo, Ohio, 1983 (Heldermann-Verlag, 1984), pp. 375–391.Google Scholar
[18]Nickolas, P., ‘Reflexivity of topological groups’, Proc. Amer. Math. Soc. 65 (1977), 137–141.CrossRefGoogle Scholar
[19]Noble, N., ‘k-Groups and duality’, Trans. Amer. Math. Soc. 151 (1970), 551–561.Google Scholar
[20]Nummela, E., ‘The completion of a topological group’, Bull. Austral. Math. Soc. 21 (1980), 407–417.CrossRefGoogle Scholar
[21]Pestov, V., ‘Some properties of free topological groups’, Moscow Univ. Math. Bull. 37 (1982), 46–49.Google Scholar
[22]Pestov, V., ‘Free topological Abelian groups and the Pontryagin duality’, Moscow Univ. Math. Bull. 41 (1986), 1–4.Google Scholar
[23]Pestov, V., ‘Universal arrows to forgetful functors from categories of topological algebra’, Bull. Austral. Math. Soc. 48 (1993), 209–249.CrossRefGoogle Scholar
[24]Rai˘kov, D.A., ‘Harmonic analysis on commutative groups with Haar measure and the theory of characters’, (in Russian), Trudy Mat. Inst. Steklov 14 (1945), 1–86.Google Scholar
[25]Ra˘kov, D.A., ‘On the completion of topological groups’, (in Russian), Izv. Akad. Nauk SSSR. Ser. Mat. 10 (1946), 513–528.Google Scholar
[26]Ra˘kov, D.A., ‘Free locally convex spaces for uniform spaces’, (in Russian), Mat. Sb. (N.S.) 63 (1964), 582–590.Google Scholar
[27]Remus, D. and Trigos-Arrieta, F.J., ‘Abelian groups which satisfy Pontryagin duality need not respect compactness’, Proc. Amer. Math. Soc. 117 (1993), 1195–1200.CrossRefGoogle Scholar
[28]Smith, M., ‘The Pontryagin duality theorem in linear spaces’, Ann. of Math. 56 (1952), 248–253.CrossRefGoogle Scholar
[29]Tkachenko, M.G., ‘On completeness of free Abelian topological groups’, Soviet Math. Dokl. 27 (1983), 341–345.Google Scholar
[30]Uspenskiiˇ, V.V., ‘On the topology of free locally convex space’, Sov. Math. Dokl. 27 (1983), 781–785.Google Scholar
[31]Wheeler, R.F., ‘Weak and pointwise compactness in the space of bounded continuous functions’, Trans. Amer. Math. Soc. 266 (1981), 515–530.CrossRefGoogle Scholar