My question is about part of the Borel Transgression Theorem on spectral sequences, translated from [1, Theorem 13.1] (see also [2] for a translated version of the whole paper):



> 
> Let $B^\bullet := \bigoplus\_{p \in \mathbb{N}\_0} B^p$ and $P^\bullet := \bigoplus\_{q=0}^m P^q$ be finite-dimensional, graded vector spaces, with $P^q$ only nonzero in odd degree.
> 
> 
> Let further $\{E\_r^{p,q}, d\_r\}\_{r \geq 0}$ be a cohomological spectral sequence whose second page has the shape
> \begin{align\*}
> E\_2^{p,q} = B^p \otimes \Lambda^q P,
> \end{align\*}
> and which converges towards a graded vector space $H^\bullet$ with $H^k = 0$ for all $ k \leq n$, where $n$ some number with $n \geq 2 m +1 $.
> 
> 
> a) Then $\Lambda^\bullet P$ is an exterior algebra of some subspace $P' \subset P$ and $P'$ admits a basis of transgressive, homogeneous elements $x\_1,\dots,x\_l$. Further, $P'$ contains all transgressive elements of $\Lambda^\bullet P$.
> 
> 
> b) If $y\_i \in B^\bullet$ fulfils $d\_r( 1 \otimes x\_i) = y\_i \otimes 1$ for $i = 1,\dots,l$, then $B^\bullet = \mathbb{K}[y\_1,\dots,y\_l]$ in degree $\leq n$.
> 
> 
> 


**Question:** Can one still say anything when $H^k = 0$ in a smaller range of $k$, like $k \leq m$ only?




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The reason I am asking is because Gelfand and Fuks use this theorem to analyze a Hochschild-Serre spectral sequence for the Lie algebra of formal vector fields on $\mathbb{R}^n$, denoted $W\_n$, see [3, Proposition 5.1], but it does not fulfill the requirements at all: In this paper, $H^k = H^k(W\_n)$ is only known to be zero for $1 \leq k \leq n$, but $P$ has one generator in every dimension $1,3,\dots,2 n - 1$, so $m = 2n - 1$.


There are some arguments in this paper that by relating the spectral sequences for different dimensions $n$, one still gets transgressive generators, but since *none* of the spectral sequences fulfill the requirements, I am not sure how Borel's theorem can be applied at all. Are there clever cutoff tricks that can salvage this?


[1] *Borel, Armand*, [**Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts**](http://dx.doi.org/10.2307/1969728), Ann. Math. (2) 57, 115-207 (1953). [ZBL0052.40001](https://zbmath.org/?q=an:0052.40001).


[2] *Novikov, S. P. (ed.); Taimanov, I. A. (ed.)*, Topological library. Part 3: Spectral sequences in topology. Transl. by V. P. Golubyatnikov, Series on Knot and Everything 50. Hackensack, NJ: World Scientific (ISBN 978-981-4401-30-2/hbk). ix, 576 p. (2012). [ZBL1264.55002](https://zbmath.org/?q=an:1264.55002).


[3] *Gel’fand, I. M.; Fuks, D. B.*, [**Cohomology of the Lie algebra of formal vector fields**](http://dx.doi.org/10.1070/IM1970v004n02ABEH000908), Math. USSR, Izv. 4(1970), 327-342 (1971); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 34, 322-337 (1970). [ZBL0216.20302](https://zbmath.org/?q=an:0216.20302).



Borel's transgression theorem for spectral sequences
Mimura and Toda's statement in *The Topology of Lie Groups*, Theorem VII.2.9 (p. 378), requires less severe degree constraints. They start with a Serre fibration $F \to E \to B$ with $B$ simply-connected, and demand only that


* there be an oddly-graded space $P$ of "generators" of degrees no more than some fixed $N$ such that the induced additive map $\Lambda P \to H^\*(F;K)$ is bijective in degrees $\leq N$ and injective in degree $N+1$ and
* $\tilde{H}{}^\*(E;K)$ be zero in degrees $\leq N + 2$.


They conclude then that transgressive elements $x\_i$ of $H^\*(F;K)$ (generating it up to degree $N$ and such that the associated map $\Lambda[x\_i] \to H^\*(F;K)$ is injective in degree $N+1$) can be chosen such that lifts $y\_i \in H^\*(B;K)$ of their transgressions are generators in the sense that the associated ring map $K[y\_i] \to H^\*(B;K)$ is bijective in degrees $\leq N+1$ and injective in degree $N+2$.


I do not believe their proof uses anything about the Serre spectral sequence beyond the fact it is a spectral sequence of differential graded algebras whose $E\_2$ page is of the form $H^\*(B;K) \otimes H^\*(F;K)$ (when $K$ is a field), so it should give a purely algebraic result about spectral sequences with $E\_2$ pages of this form.


