# Borel's transgression theorem for spectral sequences

My question is about part of the Borel Transgression Theorem on spectral sequences, translated from [1, Theorem 13.1] (see also  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?

 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.

 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.

 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.

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.