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.

  • $\begingroup$ A remark is that if $M = S^n$ is a sphere, then the spectral sequence collapses if and only if $n$ is odd, and the fibration has a cross-section if and only if $n$ is odd. $\endgroup$ Aug 7, 2018 at 11:20
  • $\begingroup$ @DanPetersen True for $k \geq 3$. $\endgroup$
    – Cihan
    Aug 7, 2018 at 18:45

1 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.