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How to use Serre spectral sequence to compute cup product structures?

Let $F\to E\to B$ be a fibration. Suppose all the differentials of the corresponding Serre spectral sequence of cohomology are zero. Can we obtain that for any field $k$, $$ H^*(E;k)\cong H^*(F;k)\otimes H^*(B,k) $$ as rings?

For example, how to solve the following problem?

enter image description here

How to use Serre spectral sequence to compute cup product structures? For example, how to solve the following problem?

enter image description here

How to use Serre spectral sequence to compute cup product structures?

Let $F\to E\to B$ be a fibration. Suppose all the differentials of the corresponding Serre spectral sequence of cohomology are zero. Can we obtain that for any field $k$, $$ H^*(E;k)\cong H^*(F;k)\otimes H^*(B,k) $$ as rings?

For example, how to solve the following problem?

enter image description here

Source Link
QSR
QSR
  • 2.2k
  • 10
  • 26

Cup product of cohomology in a Serre spectral sequence

How to use Serre spectral sequence to compute cup product structures? For example, how to solve the following problem?

enter image description here