Does the Serre spectral sequence of the Fadell-Neuwirth fibration collapse if there is a cross-section?

Question

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.

Answer 1

Nice question! Here is a partial answer which was too long for a comment.

A necessary condition for collapse at $E_2$ is that the homomorphism $i^*: H^*(\operatorname{PConf}_k(M))\to H^*(\operatorname{PConf}_{k-1}(M-\{{\rm pt}\}))$ induced by fibre inclusion is an epimorphism. This is seen by identifying this homorphism with the edge homomorphism of the spectral sequence.

If we now assume that $H^*(M)$ is torsion-free, and that the system of local coefficients on $M$ is trivial, so that $$E_2^{p,q}\cong H^p(M)\otimes H^q( \operatorname{PConf}_{k-1}(M-\{{\rm pt}\})),$$ then this necessary condition is also sufficient. This follows from the multiplicative nature of the spectral sequence. A reference is McCleary's book on spectral sequences, Theorems 5.9 and 5.10.

Now I claim that when $k=2$ and $M$ is an oriented closed manifold with torsion-free cohomology, the spectral sequence always collapses at $E_2$ (regardless of whether the first Fadell-Neuwirth fibration admits a section). Let $\Delta\in H^{\dim M}(M\times M)$ denote the diagonal class (i.e., the Poincaré dual of the diagonal submanifold $d: M\to M\times M$). Then a theorem of Cohen and Taylor gives an algebra isomorphism $$ H^*(\operatorname{PConf}_2(M))\cong \frac{H^*(M\times M)}{(\Delta)}. $$ (This can be seen using the long exact sequence of the pair $(M\times M,\operatorname{PConf}_2(M))$, together with excision and the Thom isomorphism for the normal bundle of the diagonal.) On the other hand, letting $\mu\in H^{\dim M}(M)$ denote the fundamental class (dual to $1\in H^0(M)$), we get an algebra isomorphism $$ H^*(M-\{{\rm pt}\})\cong \frac{H^*(M)}{(\mu)}. $$ The fibre inclusion homomorphism $i^*:H^*(\operatorname{PConf}_2(M))\to H^*(M-\{{\rm pt}\})$ is induced by the homomorphism $H^*(M\times M)\to H^*(M)$ which sends $1\times x$ to $x$, and is therefore surjective.

I suspect the answer to your question to be no in general, but I don't have a good enough understanding of the differentials to produce an example. Perhaps the following paper may be helpful in this regard:

Shih, Weishu, Homologie des espaces fibrés, Publ. Math., Inst. Hautes Étud. Sci. 13, 93-176 (1962). ZBL0105.16903.