Seminar Sophus Lie 3 (1993) 69–70

The solution of the epimorphism problem for Hausdorff topological groups

Vladimir Uspenskiĭ

Consider the category G of Hausdorff topological groups. Then a morphism e: G → H in G is an epimorphism if for any pair of morphisms f, g: H → K satisfying fe = ge we must have f = g. Clearly every morphism e with \overline{e(G)} = H is an epimorphism in G. It was an open problem since the late sixties whether the converse is true. This problem is equivalent to the following problem in [3]: Problem 512. [K. H. Hofmann] If G is a Hausdorff topological group and H a proper closed subgroup of G, must there exist distinct morphisms of Hausdorff topological groups f, g: G → K with f|H = g|H?

There is a considerable literature on the problem with many partial results (see the references below). On the power of these partial results it was conjectured that the answer to the problem is yes. However, the answer is negative.

Theorem 1. Let 𝕋 be a circle (a one-dimensional torus). Let G = Aut 𝕋 be the group of all self-homeomorphisms of 𝕋 with the topology of uniform convergence, and let H be the stability subgroup at some point in 𝕋. Then for any pair of morphisms f, g: G → K into a Hausdorff topological group with f|H = g|H we must have f = g.

Proof. (Sketch) Define j: G × G → K by j(x, y) = f(x)g(y)^{−1} . Let U be the coarsest uniformity on G × G for which j is uniformly continuous w.r.t. inf{L, R} (where L and R denote the left- and right- uniform structures on K, respectively). Let H act on the right on G × G via (x, y)·h = (xh, yh), and denote the orbit space of this action by X. Equip X with the quotient uniformity of (G × G, U). Since j(xh, yh) = j(x, y) for all h ∈ H, the map j induces a map j′: X → K. It turns out that the image Δ of diag(G × G) in X considered as a uniform subspace of X has the trivial (coarsest) uniformity. Since K is Hausdorff, j′ is constant on Δ which means that j is constant on diag(G × G). This says f = g.

A detailed proof will appear elsewhere.

Acknowledgement. This note was prepared while the author was visiting the Technische Hochschule Darmstadt. He thanks K. H. Hofmann cordially for his hospitality and for his assistance in preparing this announcement.

References

[1] LaMartin, F., Epics in the category of T_{2} -k -groups need not have dense range, Colloq. Math. 36 (1976), 37–41.

[2] —, On the foundations of k -group theory, Dissertationes Math. CXLVI (1977).

[3] Mill, J. van, and Reed, G., Eds., “Open Problems in Topology”, North Holland 1990.

[4] Nickolas, P., Free products of topological groups with a closed subgroup amalgamated, J. Austr. Math. Soc. 40 (1986), 414–420.

[5] Nummela, E. On epimorphisms of topological groups, General Topology and Appl. 9 (1978), 155–167.

[6] Poguntke, D., Epimorphisms of compact groups are onto, Proc. Amer. Math. Soc. 26 (1970), 503–504.

[7] Smith-Thomas, B. V., Do epimorphisms of Hausdorff groups have dense range? Notices Amer. Math. Soc. 20 (1973), A-99.

[8] —, Categories of topological groups, Quaestiones Math. 2 (1977), 355– 377.

Mathematisches Institut Universität München Theresienstraße 39 D-8000 München 2, Germany uspenskij@mathematik.uni-muenchen.dbp.de

Received May 27, 1993