**Mapping cone**

Let be a map in . We construct the mapping cone , where is identified with for all .

Proposition:

For any map we have if and only if has an extension to .

Proof:

By an earlier proposition (2.32 in \cite{Switzer2002}), iff has an extension .

() If , define by , . Note that . Since induces a map which satisfies . That is .

() If has an extension , then define by . We have . Then That is, .

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