# Cofibration

A triple $( X, i, Y)$, where $X, Y$ are topological spaces and $i: X \rightarrow Y$ is an imbedding with the following property, known as the homotopy extension property with respect to polyhedra: For any polyhedron $K$, any mapping $f: Y \rightarrow K$ and any homotopy

$$F: X \times [ 0, 1] \rightarrow K$$

with

$$F\mid _ {X \times \{ 0 \} } = f \circ i$$

there exists a homotopy

$$G: Y \times [ 0, 1] \rightarrow K$$

such that

$$G\mid _ {Y \times \{ 0 \} } = f \ \ \textrm{ and } \ \ G \circ ( i \times \mathop{\rm id} ) = F,$$

where

$$( i \times \mathop{\rm id} ): X \times [ 0, 1] \rightarrow Y \times [ 0, 1].$$

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

$$X \rightarrow Y \rightarrow Y/i ( X) \rightarrow C _ {1} \rightarrow C _ {2} \rightarrow \dots$$

of topological spaces in which $C _ {1} \sim SX$( $SX$ is the suspension of $X$) is the cofibre of the mapping $Y \rightarrow Y/i ( X)$— being converted into a cofibration, $C _ {2} \sim SY$ is the cofibre of the mapping $Y/i ( X) \rightarrow C _ {1}$, etc. If $( X, i, Y)$ is a cofibration of pointed spaces, then for any pointed polyhedron $K$ the induced sequence

$$[ X, K] \leftarrow [ Y, K] \leftarrow [ Y/i ( X), K] \leftarrow [ C _ {1} , K] \leftarrow \dots$$

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)