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[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 $H\sp*(E;k)\to H\sp*(F;k)$ is surjective. This is equivalent to saying that $\pi\sb 1(B)$ acts trivially on $H\sp*(F;k)$ and the Serre spectral sequence collapses at $E\sp 2$. S. Halperin conjectured that for $char(k)=0$ and F a 1-connected rationally elliptic space (i.e., both $H\sp*(F;{\mathcal{Q}})$ and $\pi\sb*(F)\otimes {\mathcal{Q}}$ are finite dimensional) such that $H\sp*(F;k)$ 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 $H\sp*(C\sp*(\pi\sb*(\Omega F)\otimes {\mathcal{Q}}))$ is finite dimensional. (ii) F is a rationally coformal space. (iii) The cohomology algebra $H\sp*(F;k)$ has a presentation $k[x\sb 1,...,x\sb n]/(f\sb 1,...,f\sb m)$ in which! (English) |
