[For the entire collection see Zbl 0699.00032.] A fibration $F\to E\to B$ is called totally noncohomologuous to zero (TNCZ) with respect to the coefficient field k, if $H^{*}(E;k)\to H^{*}(F;k)$ is surjective. This is equivalent to saying that ${\pi }_1\left(B\right)$ acts trivially on $H^{*}(F;k)$ and the Serre spectral sequence collapses at $E^2$. S. Halperin conjectured that for $char\left(k\right)=0$ and F a 1-connected rationally elliptic space (i.e., both $H^{*}(F;𝒬)$ and ${\pi }_{*}\left(F\right)\otimes 𝒬$ are finite dimensional) such that $H^{*}(F;k)$ vanishes in odd degrees, every fibration $F\to E\to B$ is TNCZ. The author proves this being the case under either of the following additional hypotheses: (i) The Lie algebra cohomology $H^{*}\left(C^{*}({\pi }_{*}\left(\Omega F\right)\otimes 𝒬)\right)$ is finite dimensional. (ii) F is a rationally coformal space. (iii) The cohomology algebra $H^{*}(F;k)$ has a presentation $k[x_1,...,x_n]/(f_1,...,f_m)$ in which!

EUDML-ID : urn:eudml:doc:220154
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@article{701468,
     title = {Towards one conjecture on collapsing of the Serre spectral sequence},
     booktitle = {Proceedings of the Winter School "Geometry and Physics"},
     series = {GDML\_Books},
     publisher = {Circolo Matematico di Palermo},
     address = {Palermo},
     year = {1990},
     pages = {[151]-159},
     mrnumber = {MR1061796},
     zbl = {0705.55007},
     url = {http://dml.mathdoc.fr/item/701468}
}

Markl, Martin. Towards one conjecture on collapsing of the Serre spectral sequence, dans Proceedings of the Winter School "Geometry and Physics", GDML_Books,  (1990), pp. [151]-159. http://gdmltest.u-ga.fr/item/701468/