# 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 $X \times [0,1] \sqcup Y \sqcup Z$ 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

## 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.