# Double mapping cylinder

From Topospaces

## Definition

Suppose are topological spaces and and are continuous maps. The **double mapping cylinder** of and is defined as the quotient of via the relations and .

## Particular cases

- Mapping cylinder: Here and is the identity map
- Mapping cone: Here is a one-point space and is the map to that one point
- Join: The join of spaces and is the double mapping cylinder where , , and the maps are simply projections onto the coordinates

## Generalizations

## Facts

There is a relation between the homology of the double mapping cylinder of and , and the homologies of the spaces , and . The relation is described by the exact sequence for double mapping cylinder.