Mapping cylinder

cylindrical construction

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.

References

[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)

Comments

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

References

[a1]	G.W. Whitehead, "Elements of homotopy theory" , Springer (1978) pp. 22, 23