Hatcher, *Algebraic Topology*, Chapter 0

**11. Show that is a homotopy equivalence if there exist maps such that and . More generally, show that is a homotopy equivalence if and are homotopy equivalences.**

*Proof.* Suppose that is a map and that there exist maps for which and . The fact that is a homotopy equivalence (i.e., that it has a homotopy inverse) follows in numerous ways from these hypotheses. Here are a few.

- Because , left composition with yields that and that this is homotopic to . Hence, . In particular, then, , which combined with the fact that yields the result.
- Because , right composition with similarly yields that , whereby it follows that , which combined with yields the result again.

Note that this is hardly the only two ways to deduce the same claim and, indeed, crazy combinations of compositions yield the same thing. For example implies that \implies that implies that . Tracing back through, this implies that and hence that .

To show the more general result, suppose that and are homotopy equivalences with homotopy inverses and , respectively. Then:

- By definition, so the obvious candidate for a homotopy inverse here is . It suffices to show that . To that end, note that , whereby left composition by implies that . In particular, then, , and so the result holds.
- Proceed as above: yields a candidate , and because , right composition by shows that . Hence, and the result again holds.

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You need injectivity and/or surjectivity assumptions to use “cancellation laws” for composition of functions.

Indeed, you are correct. Thanks for pointing out my absentmindedness!

Thanks for your proof. Here is some typing error: Paragraph 4 the formula in last sentence “this implies that $fhgh \simeq 1_Y \simeq fg$” should be changed to $fhgh \simeq 1_Y \simeq fh$

Thanks for your comment!

Ignoring the fact that was pointed out above (namely, that all of my ‘cancellations’ were nonsense as written), I think your comment may have a typo and that my entry may have

tonsof typos. ^_^The last sentence of that paragraph says: “…this implies that and hence that .”

And, looking back through: I feel like my entire entry is malformed. I’m not really in Hatcher mode (and haven’t been for some time) so it’s unlikely I’ll even attempt to rectify one/any/all of the changes needed here for (at least) quite some time. As it stands, though, it’s very possible that there are typos abound, so I would suggest you proceed with (at worst) caution (and at best complete disregard for this entry’s existence) before attempting to reconcile anything here whatsoever.

Thanks for reading, and for taking the time to comment.