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

Hilton, Peter; Roitberg, Joseph

Proc. Am. Math. Soc. 90, 316-320 (1984).

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

Keywords:

monomorphism; co-Hopfian objects; homotopy category; path-connected CW- spaces; epimorphism; fundamental group

References:

[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 · doi:10.1007/BF01194830
[4]	H. Hopf, Beiträge zur Klassifizierung der Flächenabbildungen, J. Reine Angew. Math. 165 (1931), 225-236. · JFM 57.0726.01