# Mapping cylinder

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 constructioncylindrical 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 |

**How to Cite This Entry:**

Mapping cylinder.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Mapping_cylinder&oldid=18232