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 if

is a homotopy between the two maps,

is a map over B for 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.

The following proof is based on the proof of Proposition in Ch.

for a homotopy over B.

We first note that it is enough to show that ƒ admits a left homotopy inverse over B.

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

Now, since ƒ is a homotopy equivalence, it has a homotopy inverse g. Since

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 gƒ, 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

be a homotopy from ƒ to

and since p is a fibration, the homotopy

lifts to a homotopy

Note also

We show

is a left homotopy inverse of ƒ over B.

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: