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X is a two-sided homotopy inverse of f relative to { M }. 12. The mapping cylinder Let /: X -± Y be a given map. By the process of topological identification, we shall construct a space Mf which is called the mapping cylinder off. For this purpose, let us consider the topological sum W (X x /) U of the spaces X x / and Y.

Then our most general problem can be displayed diagrammatically by x----f---÷ Y \ g\z/h/ Z where the maps are understood to be relative to { M }, / and one of g, h are given, and the other is to be found such that hg c_—_ f rel { M }. Nearly everything done in this chapter for the simplest extension, deformation and lifting problems admits generalizations some of which will be given in the exercises. Although the problem is formulated in the most general form, the only cases considered in remainder of the book have 0, 1 or 2 elements as the domain M of the index it.

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