I had asked a vague question in MSE where a useful pointer to the Leray-Hirsch theorem was mentioned by Mike Miller 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

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

Is there an $M$ and $k$ such that $f_k$ has a cross-section $s \colon M \rightarrow \text{PConf}_k(M)$ but the spectral sequence does not collapse on the $E_2$ page?

Remark 1: There are 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, Corollary 5.1). This suggests many cases to look at but I haven't seen an explicit computation addressing the issue.

I had asked a vague question in MSE where a useful pointer to the Leray-Hirsch theorem was mentioned by Mike Miller 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 are 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, Corollary 5.1). This suggests many cases to look at but I haven't seen an explicit computation addressing the issue.