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