Mapping cylinder: Difference between revisions

Revision as of 23:33, 2 November 2007

Definition

Let $f : X \to Y$ be a function. Then the mapping cylinder of $f$ is defined as the quotient of the disjoint union of $X \times I$ with $Y$ , modulo the equivalence relation:

$(x, 1) ≃ f (x)$

Facts

The significance of the mapping cylinder is that it is homotopy-equivalent to $Y$ , and moreover the inclusion of $X$ (say via $x \mapsto (x, 0)$ ) in the mapping cylinder is equivalent to the map $f$ .

Thus, starting from an arbitrary continuous map, we have got a homotopy-equivalent map which is an inclusion.

Further, the inclusion of $X$ in the mapping cylinder is a cofibration, which makes it even nicer.

@@ Line 4: / Line 4: @@
 <math>(x,1) \simeq f(x)</math>
+==Facts==
+The significance of the mapping cylinder is that it is homotopy-equivalent to <math>Y</math>, and moreover the inclusion of <math>X</math> (say via <math>x \mapsto (x,0)</math>) in the mapping cylinder is equivalent to the map <math>f</math>.
+Thus, starting from an arbitrary continuous map, we have got a homotopy-equivalent map which is an inclusion.
+Further, the inclusion of <math>X</math> in the mapping cylinder is a [[cofibration]], which makes it even nicer.