[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...

In rational homotopy theory, we show how the homotopy notion of pure fibration arises in a natural way. It can be proved that some fibrations, with homogeneous spaces as fibre are pure fibrations. Consequences of these results on the operation of a Lie group and the existence of Serre fibrations are given. We also examine various measures of rational triviality for a fibration and compare them with and whithout the hypothesis of pure fibration.

The main result of this brief note asserts, incorrectly, that there exists a rational fibration $S^2\to E\to ℂP^3$ whose total space admits nonzero Massey products. The methods used would be appropriate for showing results of this kind, if the circumstances were to allow for it. Unfortunately the author makes a simple, but nonetheless fatal, computational error in his calculation that ostensibly shows the existence of a nonzero Massey product (p. 249, 1.13: $ab\ne D\left(x^{2}y\right))$. In fact, for any rational fibration $S^2\to E\to ℂP^3$ the...

In the rational cohomology of a 1-connected space a structure of $C_{\infty }$-algebra is constructed and it is shown that this object determines the rational homotopy type.

Sullivan associated a uniquely determined $DGA{|}_{\mathbf{Q}}$ to any simply connected simplicial complex $E$. This algebra (called minimal model) contains the total (and exactly) rational homotopy information of the space $E$. In case $E$ is the total space of a principal $G$-bundle, ($G$ is a compact connected Lie-group) we associate a $G$-equivariant model $U_G\left[E\right]$, which is a collection of “$G$-homotopic” $DGA$’s${|}_{\mathbf{R}}$ with $G$-action. $U_G\left[E\right]$ will, in general, be different from the Sullivan’s minimal model of the space $E$. $U_G\left[E\right]$ contains the...