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?

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**, Ann. Math. (2) 57, 115-207 (1953). ZBL0052.40001.

[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.

[3] *Gel’fand, I. M.; Fuks, D. B.*, **Cohomology of the Lie algebra of formal vector fields**, Math. USSR, Izv. 4(1970), 327-342 (1971); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 34, 322-337 (1970). ZBL0216.20302.