Epimorphisms in Topological Groups
by
Stephen Watson
York University

A continuous one-to-one homomorphism f: G --> H is said to be an epimorphism if, for any K, and for any g, h: H --> K, gf = hf implies g = h. We show that there is a group G of cardinality c and, for each cardinal kappa, an epimorphic embedding of G into a group of cardinality at least kappa. This solves the Dikranjan-Tholen cowellpoweredness problem. We show that there is an epimorphic embedding of a group H into a group G and a closed normal subgroup K of H such that there is no normal subgroup F of G such that F \cap H = G. More, we show that, for all finite groups F, there is a epimorphic embedding p of some H into some G with a continuous homomorphism f: H --> F but no g: G --> F such that f = gp. We do not know what happens for arbitrary F. We show that there is an increasing chain of groups G₁, G₂, ... such that each G_n is a closed subgroup of finite index in G_n+1 and, letting G be the direct limit of the G_n's, the embedding of each G_n into G is an epimorphic embedding. Some of these results have been obtained in joint work with Guillaume Brummer.

Date received: February 28, 1997

Copyright © 1997 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caam-30.