# Fiber-homotopy equivalence

In algebraic topology, a **fiber-homotopy equivalence** is a map over a space *B* that has homotopy inverse over *B* (that is we require a homotopy be a map over *B* for each time *t*.) It is a relative analog of a homotopy equivalence between spaces.

Given maps *p*:*D*→*B*, *q*:*E*→*B*, if ƒ:*D*→*E* is a fiber-homotopy equivalence, then for any *b* in *B* the restriction

is a homotopy equivalence. If *p*, *q* are fibrations, this is always the case for homotopy equivalences by the next proposition.

**Proposition** — Let be fibrations. Then a map over *B* is a homotopy equivalence if and only if it is a fiber-homotopy equivalence.

## Proof of the proposition

The following proof is based on the proof of Proposition in Ch. 6, § 5 of (May). We write for a homotopy over *B*.

We first note that it is enough to show that ƒ admits a left homotopy inverse over *B*. Indeed, if with *g* over *B*, then *g* is in particular a homotopy equivalence. Thus, *g* also admits a left homotopy inverse *h* over *B* and then formally we have ; that is, .

Now, since ƒ is a homotopy equivalence, it has a homotopy inverse *g*. Since , we have: . Since *p* is a fibration, the homotopy lifts to a homotopy from *g* to, say, *g'* that satisfies . Thus, we can assume *g* is over *B*. Then it suffices to show *g*ƒ, which is now over *B*, has a left homotopy inverse over *B* since that would imply that ƒ has such a left inverse.

Therefore, the proof reduces to the situation where ƒ:*D*→*D* is over *B* via *p* and . Let be a homotopy from ƒ to . Then, since and since *p* is a fibration, the homotopy lifts to a homotopy ; explicitly, we have . Note also is over *B*.

We show is a left homotopy inverse of ƒ over *B*. Let be the homotopy given as the composition of homotopies . Then we can find a homotopy *K* from the homotopy *pJ* to the constant homotopy . Since *p* is a fibration, we can lift *K* to, say, *L*. We can finish by going around the edge corresponding to *J*:

## References

- May, J.P. A Concise Course in Algebraic Topology, (1999) Chicago Lectures in Mathematics ISBN 0-226-51183-9
*(See chapter 6.)*