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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

MSC:

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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References:

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[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
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[4] H. Hopf, Beiträge zur Klassifizierung der Flächenabbildungen, J. Reine Angew. Math. 165 (1931), 225-236. · JFM 57.0726.01
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