Computation of total space of fibration using Leray-Serre spectral sequence

Question

If a Leray-Serre spectral sequence corresponding to fibration $F\hookrightarrow E\to B$ with $E_2^{2,t}=H^s(B;k)\otimes H^t(F;k)$ converges to $H^{*}(E;k)$ as an algebra then the generators of graded algebra $H^{*}(E;k)$ comes only from edge homomorphisms? Can $E^{p,q}_{\infty},p,q\not=0$ give a generator of $H^{*}(E;k)$?

Answer 1

Suppose there are classes $b_1 \in E_2^{1,0}$, $b_2 \in E_2^{2,0}$ and $f \in E_2^{0,1}$, with $b_1 b_2 = 0$. Suppose further that there is a differential $d_2: f \mapsto b_2$. Then the class $b_1 \otimes f$ will survive and be an algebra generator.