I had asked a vague [question][1] in MSE where a useful pointer to the Leray-Hirsch theorem was mentioned by [Mike Miller][2] in the comments, but received no answers. Here I will specialize to an interesting case to get a well-posed question.

Let $M$ be a closed manifold of dimension $\geq 2$ and consider the Fadell-Neuwirth fibration $f_k\colon \text{PConf}_k(M) \rightarrow M$ where $\text{PConf}_k(M)$ is the ordered configuration space ($f_k$ forgets all but one point) for some $k \geq 2$. The associated cohomology Serre spectral sequence has the form

$$E_2^{p,q} = H^p(M ; H^q(\text{PConf}_{k-1}(M - \{pt\}) ; \mathbb{Z})) \Rightarrow H^{p+q}(\text{PConf}_k(M))$$ 

> If $f_k$ has a cross-section $s \colon
 M \rightarrow \text{PConf}_k(M)$, does the spectral sequence
> collapse on the $E_2$ page?

**Remark 1:** [There][4] [are][3] Serre fibrations with a cross-section that do not exhibit a collapse on $E_2$.

**Remark 2**: There is always a section of $f_k$, provided that $M$ has a positive first Betti number (seems to be a somewhat forgotten fact from the original Fadell-Neuwirth [paper][5], Corollary 5.1). This suggests many cases to look at but I haven't seen an explicit computation addressing the issue.


  [1]: https://math.stackexchange.com/q/2834480/26141
  [2]: https://mathoverflow.net/users/40804/mike-miller
  [3]: https://www.sciencedirect.com/science/article/pii/S0166864107003197
  [4]: https://www.projecteuclid.org/download/pdf_1/euclid.ijm/1256049512
  [5]: https://gdz.sub.uni-goettingen.de/id/PPN35397434X_0010?tify={%22pages%22:[116],%22view%22:%22info%22}