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The epimorphism problem for Hausdorff topological groups. (English) Zbl 0810.22002

Let \(\mathcal G\) be the category of Hausdorff topological groups and continuous homomorphisms. The problem in question, circulated broadly in the 1960’s by K. H. Hofmann, is this: Given \(G\) in \(\mathcal G\) and a proper closed subgroup \(H\) of \(G\), must there exist \(K\) in \(\mathcal G\) and distinct morphisms \(f\) and \(g\) from \(G\) into \(K\) such that \(f| H =g| H\)? In categorical language: Do all epimorphisms in \(\mathcal G\) have dense range? Since affirmative solutions to Hofmann’s question have been obtained in several important special cases – by A. G. Kurosh, A. K. Livshits and E. G. Schul’geifer (1960) when all groups are discrete, by D. Poguntke (1970) when all are compact, by E. Nummela (1978) when all are locally compact, by W. F. LaMartin (1979) and P. Nicholas (1986) when all are \(k_ \omega\)-groups – it is surprising that the answer is “No”. The author’s example, verified here in detail, takes for \(G\) the automorphism group of a compact connected manifold \(M\) without boundary in the topology of uniform convergence and for \(H\) the stability subgroup at some point of \(M\).
Given \(G_ i\) in \(\mathcal G\) (\(i = 0,1\)) with a common closed subgroup \(A\), the so-called free product \(P = G_ 0 *_ A G_ 1\) in \(\mathcal G\) with \(A\) amalgamated is characterized as follows: (a) \(P\) is an object of \(\mathcal G\); (b) there are canonical morphisms \(g_ i : G_ i \to P\) in \(\mathcal G\) which agree on \(A\), and (c) for every pair of morphisms \(h_ i : G_ i \to Q\) in \(\mathcal G\) which agree on \(A\) there is a unique morphism \(f : P \to Q\) such that \(h_ i = fg_ i\). The author notes, thus settling another question closely related to Hofmann’s, that his group \(G\) and subgroup \(H\) have this property: the free product \(G *_ H G\) in \(\mathcal G\) with \(H\) amalgamated coincides with \(G\) itself, hence is not equal (considered as a group without topology) to the free product \(G *_ H G\) in the category of groups without topology. It remains unsolved, however, whether in \(\mathcal G\) the canonical morphisms \(g_ i : G_ i \to P = G_ 0 *_ A G_ 1\) must always be homeomorphic embeddings.

MSC:

22A05 Structure of general topological groups
08B25 Products, amalgamated products, and other kinds of limits and colimits
54H11 Topological groups (topological aspects)
18B40 Groupoids, semigroupoids, semigroups, groups (viewed as categories)
54E15 Uniform structures and generalizations
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References:

[1] Comfort, W. W., Problems on topological groups and other homogeneous spaces, (van Mill, J.; Reed, G. M., Open Problems in Topology (1990), North-Holland: North-Holland Amsterdam), 315-347 · Zbl 1201.22001
[2] Dierolf, S.; Schwanengel, U., Examples of locally compact non-compact minimal topological groups, Pacific J. Math., 82, 349-355 (1979) · Zbl 0388.22002
[3] Fay, T. H., A note on Hausdorff groups, Bull. Austral. Math. Soc., 13, 117-119 (1975) · Zbl 0302.18001
[4] Katz, E.; Morris, S. A., Free products of topological groups with amalgation II, Pacific J. Math., 120, 123-130 (1985) · Zbl 0589.22001
[5] Khan, M. S.; Morris, S. A., Free products of topological groups with central amalgamation II, Trans. Amer. Math. Soc., 273, 405-432 (1982) · Zbl 0496.22004
[6] LaMartin, W. F., Epics in the category of \(T_2-k\)-groups need not have dense range, Colloq. Math., 36, 37-41 (1976) · Zbl 0353.22001
[7] LaMartin, W. F., On the foundations of \(k\)-group theory, Dissertationes Math., 146 (1977) · Zbl 0394.22001
[8] Ludescher, H.; de Vries, J., A sufficient condition for the existence of a \(G\)-compactification, Indag. Math., 42, 263-268 (1980) · Zbl 0447.54053
[9] Nickolas, P., Free products of topological groups with a closed subgroup amalgamated, J. Austral. Math. Soc., 40, 414-420 (1986) · Zbl 0615.22001
[10] Nummela, E., On epimorphisms of topological groups, Gen. Topology Appl., 9, 155-167 (1978) · Zbl 0388.18001
[11] Ordman, E. T., Free products of topological groups with equal uniformities I, Colloq. Math., 31, 37-43 (1974) · Zbl 0261.22001
[12] Poguntke, D., Epimorphisms of compact groups are onto, Proc. Amer. Math. Soc., 26, 503-504 (1970) · Zbl 0204.35403
[13] Roelcke, W.; Dierolf, S., Uniform Structures on Topological Groups and their Quotients (1981), McGraw-Hill: McGraw-Hill New York · Zbl 0489.22001
[14] Smith-Thomas, B. V., Do epimorphisms of Hausdorff groups have dense range?, Notices Amer. Math. Soc., 20, A-99 (1970)
[15] Smith-Thomas, B. V., Categories of topological groups, Quaestiones Math., 2, 355-377 (1977) · Zbl 0399.22003
[16] Uspenskij, V., The solution of the epimorphism problem for Hausdorff topological groups, Sem. Sophus Lie, 3, 69-70 (1993) · Zbl 0787.22003
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