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
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