Some Recent Results on Hopficity, Co-hopficity and Related Properties

Abstract

The author surveys a number of results concerning Hopfian, coHopfian, and related objects in various concrete categories. Herean object is called Hopfian (co Hopfian) provided everysurjective (injective) endomorphism is an automorphism; the namerefers back 70 years to results of Hopf on fundamental groups ofsurfaces. In particular, Hopfian and co Hopfian groups, rings,modules and topological spaces are discussed, and known existenceand non existence results in the literature are summarized. Asecondary theme of the paper is residual finiteness, harking backto work of A. Maltsev on a question of Hopf he proved thatfinitely generated, residually finite groups are Hopfian [Rec.Math. (Mat. Sbornik) N.S. 8 (50) (1940), 405 422; MR 2, 216d].(Hopf's original question was answered in the negative by B. H.Neumann, who showed that not all finitely generated groups areHopfian [J. London Math. Soc. 25 (1950), 247 248; MR 12, 390a].)Analogs of Maltsev's result were proved for rings by J. Lewin [J.Algebra 5 (1967), 84 88; MR 34#196], and for modules by theauthor [J. Ramanujan Math. Soc. 8 (1993), no. 1 2, 29 48; MR94j:16003]. The paper concludes with a discussion of 9 openproblems.\par {For the entire collection see MR\Cite{Birkenmeier01:International:Birkhauser}[2002c:16002].}