Mapping cylinder

cylindrical construction

A construction associating with every continuous mapping of topological spaces $ f: X \rightarrow Y $ the topological space $ I _ {f} \supset Y $ that is obtained from the topological sum (disjoint union) $ X \times [ 0, 1] \amalg Y $ by the identification $ x \times \{ 1 \} = f ( x) $, $ x \in X $. The space $ I _ {f} $ is called the mapping cylinder of $ f $, the subspace $ Y $ is a deformation retract of $ I _ {f} $. The imbedding $ i: X = X \times \{ 0 \} \subset I _ {f} $ has the property that the composite $ \pi \circ i: X \rightarrow Y $ coincides with $ f $( here $ \pi $ is the natural retraction of $ I _ {f} $ onto $ Y $). The mapping $ \pi : I _ {f} \rightarrow Y $ 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 $ f: X \rightarrow Y $ 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