It is a well known result in Algebraic Topology that given a covering space $E\to B$ where the base has a CW-structure, then the total space can be given a CW-structure (see for example Theorem 8.10 of Bredon's Topology and Geometry). Moreover, the covering map is cellular with respect to these cellular structures. And if both the fibre and the base are compact, then the CW-complex of the total space is finite.
From now on let's assume all spaces are compact and therefore all CW-complexes are finite (feel free to point out when this assumption is not necessary but I am only interested in the compact case since I am working with Euler-Poincaré characteristic and integration against it). Moreover, finiteness ensures we don't have to be worried about checking extra conditions,
So my first question is:
- Is it possible to do the same with a finite simplicial complex structure on the base? (so we give to the total space a finite simplicial structure)
I think that if the following lemma is true, then the response to the above question would be affirmative:
Lemma If the base is a finite regular CW-complex, then the total space has a finite regular CW-complex structure and the covering map would be cellular.
I believe the lemma is true (at least in the compact case). A brute force proof for the the compact case could be using that regular CW-complexes are triangulable. Then use barycentric subdivision so that the simplicial structure behaves well with the covering by evenly covered open sets (compactness guarantees there are only a finite number of them so this can be done). And then do the lifting.
- It could be done in such a way that the covering map is simplicial?
My thoughts about this question is that the response is negative but I am open to hear some extra assumptions which could be enough to guarantee the result.
Now let's generalize a bit.
- Is it possible to lift a CW-structure in the case of fibre bundles? (Again all the spaces are assumed to be compact) (if needed, it can be assumed that the fibre $F$ is homotopy equivalent to a finite CW-complex).
If what I am asking is too much:
- Does the total space of a fibre bundle have the homotopy type of a compact CW-complex if the base and the fibers have?
Regarding this question I think that maybe this question and its answer could be enlightening.
In order to generalize a bit further:
Is it possible to lift a CW-structure in the case of Serre (or weak) fibrations? (Again all the spaces are assumed to be compact) (if needed, it can be assumed that the fibre $F$ is homotopy equivalent to a finite CW-complex).
Is it possible to lift a CW-structure in the case of (Hurewicz) fibrations? (Again all the spaces are assumed to be compact) (if needed, it can be assumed that the fibre $F$ is a CW-complex).
Again, if what I am asking is too much:
- Does the total space of a fibration (Serre fibrations would be enough) have the homotopy type of a compact CW-complex if the base and the fibers have?
I know the result is true if I don't ask for compactness and I assume the base is path-connected (Theorem 5.4.2 Cellular structures in topology, Fritsch).
I would be happy if references or arguments were given about any of the questions. Sorry for the long post but I thought that break the questions in different posts would not be a good idea due to the strong relation between them. Any comments would be appreciated.
