[Math] A conceptual proof that local fibrations over paracompact spaces are global fibrations

Question

I'm teaching some homotopy theory at the moment, and I'm discovering a number of things I've never before learned properly. (I guess everybody who's ever taught knows that feeling.) One of those things is that a local Hurewicz fibration is a global Hurewicz fibration if the base space is paracompact. This result goes back to Hurewicz and Huebsch, is proved in Spanier's book, and there's a paper by Dold where he compares various local and global properties of projection maps, including being a fibration. None of these proofs are what I would call conceptual. There's a lot of gluing together of "extended" lifting functions, dividing up intervals in three parts, etc etc, and they use the axiom of choice. Since paracompactness needs to get used, any possible proof will probably require some gluing, but does anyone have a more conceptual proof of the result? Has anyone taught this to students, and how was that? Is the axiom of choice a necessary input?

Answer 1

Hi Tilman, it seems as we are teaching the same course at the same time! I will do this in about three weeks. There are proofs of that theorem in the book of May and in tom Dieck's new volume. May explains the conceptual idea a bit, and tom Dieck puts it into a context, relating it to other proofs (for example the bundle homotopy theorem).

I think the underlying idea is very natural and simple. The glueing is definitly required because the theorem is a local-to-global statement and paracompactness of the base is needed since paracompact spaces are the right class of spaces for local-to-global arguments.

Here is my version of the idea. Filling all the details is technical and probably leads to the proofs that you find in the literature, but it might be helpful to know the idea (and it could be an interesting exercise for you or me to work out the details, only consulting the first section of chapter 13 of tom Diecks book). Let $p:E \to B$ be a map, $\gamma$ be a path in $B$ and $p(y)=\gamma(0)$. Consider the space $X_{y,\gamma}$ of paths $\delta:[0,1] \to E$ beginning at $y$ and covering. $p$ has the path-lifting property with respect to a point iff $X_{x,\gamma}$ is $-1$-connected (i.e. nonempty) for all $x$ and $\gamma$. If $p$ is a local fibration, then $X_{y,\gamma}$ is nonempty: If the path $\gamma$ is short (stays in one of the sets where $p$ is known to be a fibration), this is clear. Otherwise, the path goes through a finite number of these sets and you have to subdivide the intervals to see that $X_{y,\gamma}$ is nonempty. But you see why you have to subdivide the interval and this subdivision is needed for the same reason as in the proof of the excision theorem in homology (but the direction is reverse: for excision, you need to localize, here we globalize).

In order to prove the Hurewicz-Huebsch theorem, you have to produce such lifts of paths in families. Let $X \times [0,1] \to B$ be a homotopy; I like to think of it as a family of paths parametrized by $X$. For any point on $X$, you find a nonempty space of local lifts. Pick such lifts, for a locally finite cover $(U_i)$, $i\in I$, of $X$; what you want is to glue them together with a partition of unity. For that, you need homotopies on the overlaps $U_{ij}$, so you better know in advance that the space of lifts on any $U_i$ is connected. But there might be triple overlaps $U_{ijk}$ and you want to glue the three lifts on $U_{ij}$, $U_{ik}$ and $U_{jk}$. So the space of local lifts is better $1$-connected. In an obvious continuation of this pattern, you see that you need to know that the space of local lifts is $\infty$-connected (contractible). In fact, if $p$ is a fibration, then the space of lifts is contratible; you need some formal nonsense tricks and the definition of a fibration for that. Let us go back to a local fibration $E \to B$ and a path $\gamma$ in $X$. As long as your path is short, the above argument for fibrations tells you that the space of lifts is not merely $-1$- but $\infty$-connected. This is also true if the path is not short, again by subdividing the interval. Do the same argument in small families and globalize by the following construction: for any finite nonemtpy $S \subset I$, pick a map from the simplex on $S$ to the space of lifts over $U_S =\cap_{i \in S} U_i$, in a compatible way for inclusions of index sets (induction on $|S|$ and contractibility of the spaces of local lifts). Finally, use a partition of unity.

Once your students understand this idea (to construct something with a local flavour on a space, construct it locally, show the the space of local constructions is contractible, use this contractibility and partitions of unity to glue these local things together), then the bundle homotopy theorem is easy-going. In fact, if you think about the content of the bundle homotopy theorem and the theorem "fibre bundles are Hurewicz fibrations", you'll find out that they follow from each other by rather formal arguments. In my course, I will teach the bundle homotopy theorem and then "fibre bundles are Hurewicz fibrations" as a consequence, omitting the general case of the Hurewicz-Huebsch theorem.

I hope this helps and you see why all these annyoing technicalities enter. As for the axiom of choice: Im afraid I cannot help you since I never care about this :-))