# Replacing the Fibre of a Fibration

This was a question I first asked on stack exchange, here. In my head it seems like a fairly reasonable thing to ask for, but I'm not aware of any construction in the literature.

Let $$p:E\rightarrow X$$ be a Serre fibration over a pointed, connected CW complex $$X$$ with strict fibre a CW complex $$F=p^{-1}(\ast)$$. Given another space $$F'$$ and a homotopy equivalence $$F\simeq F'$$, is it possible to construct a Serre fibration $$p':E'\rightarrow X$$ with strict fibre $$F'=p'^{-1}(\ast)$$? If $$E'$$ exists, is it then possible to extend the homotopy equivalence $$F\simeq F'$$ to a fibre homotopy equivalence $$E\simeq E'$$ over $$X$$?

If you assume some extra structure on the fibration $$p$$ and the homotopy equivalence $$F\simeq F'$$ then things work out fairly easily, but I'm interested in the more general case.

Really I would like both questions to be taken together, since as pointed out in the comments, the projection $$X\times F'\rightarrow X$$ trivially satisfies the requirements of the first question alone, and such an answer is not exactly what I was looking for.

• can you please recall what "strict fiber" means? – Gael Meigniez Jul 19 '19 at 9:22
• The simplicial analogue of this is closely related to univalence, and is shown in Theorem 3.4.1 of The simplicial model of Univalent Foundations (after Voevodsky) (Kapulkin–Lumsdaine 2012). Specifically, in the triangular-prism diagram in the proof of 3.4.1, take $B$ to be your $X$, $A$ to be the point of $X$, $\bar{E_2}$ to be your $E$ (so $E_2$ is the fiber $F$), and $E_1$ to be your $F'$. The extension $\bar{E_1}$ constructed there is then your desired $E'$. I don’t think it’s quite trivial to adapt the proof to the topological setting, though. – Peter LeFanu Lumsdaine Jul 19 '19 at 9:26
• @GaelMeigniez, by 'strict fibre' I mean the inverse image $p^{-1}(\ast)\subseteq E$, with $\ast\in X$ denoting the supplied basepoint. – Tyrone Jul 19 '19 at 9:33
• For the first question you presumably want to avoid a trivial answer such as taking the projection $F'\times X \to X$. – Allen Hatcher Jul 19 '19 at 19:21

You can make the construction as follows. There are three steps.

First make a Serre fibration over the unit interval, pi:T->I=[0,1] such that the fibre over 0 (resp. 1) is F (resp. F'). You will find how to make this in the general case; for example, for F=1 point and F'=the interval, here is a proof that the triangle does it.

Consider in the plane the triangle T: 0<=y<=x<=1 and the projection pi(x,y)=x. Given a CW-complex A, a continuous map from A to T:a->(x_0(a),y_0(a)) and a homotopy (x_t(a)) (0<=t<=1), consider K={(a,t)\in AxI/x_t(a)=0} and K_0=(Ax0)\cap K. On the complement (Ax0)\K_0 you have the slope function s_0(a):=y_0(a)/x_0(a); extend it to a continuous function s from (AxI)\K to I; the wanted homotopy is (a,t)->(x_t(a),s(a,t)x_t(a)) for (a,t)\notin K and (a,t)->(0,0) for (a,t)\in K. Hence, pi is a Serre fibration.

Second, let $$f:X\rightarrow X$$ be homotopic to the identity and contract some neighborhood $$N$$ of $$*$$ onto $$*$$. Pulling back your fibration $$p$$ through $$f$$, you are reduced to the case where $$p$$ is a projection $$F\times N\rightarrow N$$ over $$N$$.

Third, consider a function $$g: N\rightarrow I$$ whose value is $$0$$ on $$\partial N$$ and $$1$$ on $$*$$; define $$E'$$ over $$N$$ as the amalgamated product of $$T$$ with $$N$$ over $$pi$$ and $$g$$; define $$E'$$ as $$E$$ over $$(X-N)$$. If you have chosen $$N$$ to retract by deformation on $$*$$, then this $$E'$$ will have the property you want.

Maybe there is a more elegant way to present the construction (i'm not an expert in homotopy theory nor cw-complexes)

• Thanks, Gael. If you work out that last step I'd be i'd be interested in seeing it. – Tyrone Jul 20 '19 at 17:52
• Gael, does the projection of a right triangle onto its base not satisfy that? – Connor Malin Jul 25 '19 at 16:10
• After Peter's comment, I have merged my partial answers. Thanks, Peter. – Gael Meigniez Jul 26 '19 at 10:52