In algebraic topology, a fiber-homotopy equivalence is a homotopy equivalence between fibers of maps into a space B from spaces D and E (that is, a map between preimages that is bidirectionally invertible up to homotopy). It is a fiber-wise 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
editThe following proof is based on the proof of Proposition in Ch. 6, § 5 of (May 1999). 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 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 . 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
edit- May, J. Peter (1999). A concise course in algebraic topology (PDF). Chicago Lectures in Mathematics. Chicago: University of Chicago Press. ISBN 0-226-51182-0. OCLC 41266205. (See chapter 6.)
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