Mapping cylinder

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A construction associating with every continuous mapping of topological spaces the topological space that is obtained from the topological sum (disjoint union) by the identification , . The space is called the mapping cylinder of , the subspace is a deformation retract of . The imbedding has the property that the composite coincides with (here is the natural retraction of onto ). The mapping is a homotopy equivalence, and from the homotopy point of view every continuous mapping can be regarded as an imbedding and even as a cofibration. A similar assertion holds for a Serre fibration. For any continuous mapping the fibre and cofibre are defined up to a homotopy equivalence.


[1] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)
[2] R.E. Mosher, M.C. Tangora, "Cohomology operations and their application in homotopy theory" , Harper & Row (1968)


The literal translation from the Russian yields the phrase "cylindrical constructioncylindrical construction" for the mapping cylinder. This phrase sometimes turns up in translations.


[a1] G.W. Whitehead, "Elements of homotopy theory" , Springer (1978) pp. 22, 23
How to Cite This Entry:
Mapping cylinder. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article