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.
|||E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)|
|||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 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|
Mapping cylinder. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Mapping_cylinder&oldid=43106