# Double mapping cylinder

This article describes a construct that involves some variant of taking a product of a topological space with the unit interval and then making some identifications, typically at the endpoints, based on some specific maps.

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## Contents

## 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 .

## More specific constructions

Specific cases of the above arise either by setting and the identity map (or correspondingly for and ) or setting or to be a one-point space. If we impose only one constraint, the resultant construction is the construction corresponding to the *other* unspecified map. If we impose two constraints, then the resulting construction depends only on the input space .

Note that the roles of and can be interchanged here.

Name of construction | One-point spaces? | Identity maps? | Remaining input? | Conclusion |
---|---|---|---|---|

mapping cylinder | -- | , so | With this stipulation, the double mapping cylinder is the same as the mapping cylinder for . | |

mapping cone | , so sends all of to one point | -- | With this stipulation, the double mapping cylinder is the same as the mapping cone for . | |

cone space | , so sends all of to one point | , so | Only the space | With this stipulation, we get the cone space for |

suspension | both and | -- | Only the space | With this stipulation, we get the suspension of |

cylinder | -- | both and | only the space | With this stipulation, we get the cylinder on , i.e., the product of and the unit interval |

### The join

The join of two spaces and can be constructed as a double mapping cylinder as follow: Set , and , and let be the coordinate projection maps.

## Generalizations

## Related notions

## 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.