Towards one conjecture on collapsing of the Serre spectral sequence

Markl, Martin

Proceedings of the Winter School "Geometry and Physics", GDML_Books, (1990), p. [151]-159 / Harvested from

Résumé

[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 $π_{1} (B)$ acts trivially on $H^{*} (F; k)$ and the Serre spectral sequence collapses at $E^{2}$ . S. Halperin conjectured that for $c h a r (k) = 0$ and F a 1-connected rationally elliptic space (i.e., both $H^{*} (F; 𝒬)$ and $π_{*} (F) \otimes 𝒬$ 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 $H^{*} (C^{*} (π_{*} (Ω F) \otimes 𝒬))$ 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!

MR MR1061796 | Zbl 0705.55007

EUDML-ID : urn:eudml:doc:220154
Mots clés:

@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/