# 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 [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.

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.