Towards one conjecture on collapsing of the Serre spectral sequence

Markl, Martin. "Towards one conjecture on collapsing of the Serre spectral sequence." Proceedings of the Winter School "Geometry and Physics". Palermo: Circolo Matematico di Palermo, 1990. [151]-159. <http://eudml.org/doc/220154>.

@inProceedings{Markl1990,
abstract = {[For the entire collection see Zbl 0699.00032.] A fibration $F\rightarrow E\rightarrow B$ is called totally noncohomologuous to zero (TNCZ) with respect to the coefficient field k, if $H^*(E;k)\rightarrow H^*(F;k)$ is surjective. This is equivalent to saying that $\pi _1(B)$ acts trivially on $H^*(F;k)$ and the Serre spectral sequence collapses at $E^2$. S. Halperin conjectured that for $char(k)=0$ and F a 1-connected rationally elliptic space (i.e., both $H^*(F;\{\mathcal \{Q\}\})$ and $\pi _*(F)\otimes \{\mathcal \{Q\}\}$ are finite dimensional) such that $H^*(F;k)$ vanishes in odd degrees, every fibration $F\rightarrow E\rightarrow B$ is TNCZ. The author proves this being the case under either of the following additional hypotheses: (i) The Lie algebra cohomology $H^*(C^*(\pi _*(\Omega F)\otimes \{\mathcal \{Q\}\}))$ 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!},
author = {Markl, Martin},
booktitle = {Proceedings of the Winter School "Geometry and Physics"},
keywords = {Geometry; Physics; Proceedings; Winter school; Srní (Czechoslovakia)},
location = {Palermo},
pages = {[151]-159},
publisher = {Circolo Matematico di Palermo},
title = {Towards one conjecture on collapsing of the Serre spectral sequence},
url = {http://eudml.org/doc/220154},
year = {1990},
}

TY - CLSWK
AU - Markl, Martin
TI - Towards one conjecture on collapsing of the Serre spectral sequence
T2 - Proceedings of the Winter School "Geometry and Physics"
PY - 1990
CY - Palermo
PB - Circolo Matematico di Palermo
SP - [151]
EP - 159
AB - [For the entire collection see Zbl 0699.00032.] A fibration $F\rightarrow E\rightarrow B$ is called totally noncohomologuous to zero (TNCZ) with respect to the coefficient field k, if $H^*(E;k)\rightarrow H^*(F;k)$ is surjective. This is equivalent to saying that $\pi _1(B)$ acts trivially on $H^*(F;k)$ and the Serre spectral sequence collapses at $E^2$. S. Halperin conjectured that for $char(k)=0$ and F a 1-connected rationally elliptic space (i.e., both $H^*(F;{\mathcal {Q}})$ and $\pi _*(F)\otimes {\mathcal {Q}}$ are finite dimensional) such that $H^*(F;k)$ vanishes in odd degrees, every fibration $F\rightarrow E\rightarrow B$ is TNCZ. The author proves this being the case under either of the following additional hypotheses: (i) The Lie algebra cohomology $H^*(C^*(\pi _*(\Omega F)\otimes {\mathcal {Q}}))$ 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!
KW - Geometry; Physics; Proceedings; Winter school; Srní (Czechoslovakia)
UR - http://eudml.org/doc/220154
ER -