Fiber-homotopy equivalence

Summary

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: DB, q: EB, if ƒ: DE 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 edit

The 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: DD 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.){{cite book}}: CS1 maint: postscript (link)