Mapping cylinder

In mathematics, the mapping cylinder of a function (mathematics) f between topological spaces X and Y is, loosely speaking, a homotopical way of considering f to be an injection of its domain into its range. More precisely, it is defined to be the topological space :M_f = ((X imes I) sqcup Y)/{sim}, where I is the unit interval and sqcup denotes the disjoint union of two topological spaces, and sim is an equivalence relation such that no two points (x,t) in X imes I are equivalent, nor two points y in Y, while we have (x,t) sim y if and only if t = 1 and f(x) = y. In short, M_f is the space obtained by pasting X into Y via f.

One use of the mapping cylinder is to apply theorems concerning subspaces or inclusions of spaces to general maps which may not be injective. This is possible since clearly X may be considered the subspace of points (x,0) in M_f, and M_f itself is homotopy equivalent to Y, such that when X is so identified with a subspace of M_f the homotopy equivalence itself reduces to f on X. This is a result of the fact that Y is in fact a retract of M_f, obtained by sending each point (x,t) on the literal cylinder over X to f(x) in Y, while fixing the points of Y. In particular, this map acts as f on the included copy of X in M_f, which validates the original claim.

Consequently, theorems or techniques (such as homology (mathematics), Cohomology, or homotopy theory itself) which are independent of the homotopy class of the spaces and maps involved may be applied to X, Y, f with the assumption that X subset Y and that f is actually the inclusion of a subspace. Another, more intuitive appeal of the construction is that it accords with the usual mental image of a function as sending points of X to points of Y, and hence of embedding X within Y, despite the fact that the function need not be one-to-one. That the construction yields a picture which is homotopy equivalent to the intuitive one indicates that intuition is a correct picture so long as deformation of Y is not an obstacle.