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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


there exists a homotopy

such that


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.


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


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

How to Cite This Entry:
Cofibration. A.F. Kharshiladze (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098