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.
Best Answer
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.