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)

Comments

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