Double mapping cylinder

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 $X\times [0,1]\bigsqcup Y\bigsqcup Z$ via the relations $(x,0)\simeq f(x)$ and $(x,1)\simeq g(x)$.

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.