Double mapping cylinder: Difference between revisions

Revision as of 23:36, 2 November 2007

Definition

Suppose $X, Y, Z$ are topological spaces and $f : X \to Z$ and $g : Y \to Z$ are continuous maps. The double mapping cylinder of $f$ and $g$ is defined as the quotient of $X \times [0, 1] ⊔ Y ⊔ Z$ via the relations $(x, 0) ≃ f (x)$ and $(x, 1) ≃ g (x)$ .

Particular cases

Mapping cylinder: Here $X = Y$ and $f$ is the identity map
Mapping cone: Here $Z$ is a one-point space and $f$ is the map to that one point
Join: The join of spaces $A$ and $B$ is the double mapping cylinder where $X = A \times B$ , $Y = A$ , $Z = B$ and the maps are simply projections onto the coordinates
Suspension: Here $Y$ and $Z$ are both one-point spaces

Generalizations

Mapping torus

Related notions

Mapping telescope

Facts

There is a relation between the homology of the double mapping cylinder of $f$ and $g$ , and the homologies of the spaces $X$ , $Y$ and $Z$ . The relation is described by the exact sequence for double mapping cylinder.

@@ Line 8: / Line 8: @@
 * [[Mapping cone]]: Here <math>Z</math> is a one-point space and <math>f</math> is the map to that one point
 * [[Join]]: The join of spaces <math>A</math> and <math>B</math> is the double mapping cylinder where <math>X = A \times B</math>, <math>Y = A</math>, <math>Z = B</math> and the maps are simply projections onto the coordinates
+* [[Suspension]]: Here <math>Y</math> and <math>Z</math> are both one-point spaces
 ==Generalizations==