Note on epimorphisms and monomorphisms in homotopy theory. (English) Zbl 0565.55009

The paper deals with Hopfian and co-Hopfian objects in the pointed homotopy category \({\mathcal K}\) of path-connected CW-spaces. An object X of a category \({\mathcal C}\) is called Hopfian if every epimorphism \(\epsilon\) : \(X\twoheadrightarrow X\) in \({\mathcal C}\) is an automorphism. In the principal theorem it is proved that Hopfian and co-Hopfian objects in \({\mathcal K}\) can be obtained under suitable finiteness conditions on homology and fundamental group assumptions.
Reviewer: K.H.Kamps


55P10 Homotopy equivalences in algebraic topology
55P30 Eckmann-Hilton duality
20F18 Nilpotent groups
55P99 Homotopy theory
55P20 Eilenberg-Mac Lane spaces
Full Text: DOI


[1] M. Castellet, P. Hilton and J. Roitberg, On pseudo-identities. II (in preparation). · Zbl 0546.20025
[2] Peter Hilton, Guido Mislin, and Joe Roitberg, Localization of nilpotent groups and spaces, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. North-Holland Mathematics Studies, No. 15; Notas de Matemática, No. 55. [Notes on Mathematics, No. 55]. · Zbl 0323.55016
[3] Peter Hilton and Joseph Roitberg, On pseudo-identities. I, Arch. Math. (Basel) 41 (1983), no. 3, 204 – 214. · Zbl 0525.20021
[4] H. Hopf, Beiträge zur Klassifizierung der Flächenabbildungen, J. Reine Angew. Math. 165 (1931), 225-236. · JFM 57.0726.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.