# Cofibration

A triple , where are topological spaces and is an imbedding with the following property, known as the homotopy extension property with respect to polyhedra: For any polyhedron , any mapping and any homotopy

with

there exists a homotopy

such that

where

If this property holds with respect to any topological space, then the cofibration is known as a Borsuk pair (in fact, the term "cofibration" is sometimes also used in the sense of "Borsuk pair" ). The space is called the cofibre of . The mapping cylinder construction converts any continuous mapping into a cofibration and makes it possible to construct a sequence

of topological spaces in which ( is the suspension of ) is the cofibre of the mapping — being converted into a cofibration, is the cofibre of the mapping , etc. If is a cofibration of pointed spaces, then for any pointed polyhedron the induced sequence

is an exact sequence of pointed sets; all terms of this sequence, from the fourth onward, are groups, and from the seventh onward — Abelian groups.

#### References

[1] | E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) |

#### Comments

In Western literature a cofibration always means what is here called a Borsuk pair.

**How to Cite This Entry:**

Cofibration.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Cofibration&oldid=13585