= Serre spectral sequence
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For a <Serre fibration> $F\to E\to B$ over a path-connected <CW complex>, the cohomological <Serre spectral sequence> starts with
$$
E_2^{p,q}=H^p(B;\mathcal H^q(F;R)),\qquad d_r:E_r^{p,q}\to E_r^{p+r,q-r+1}.
$$
The fibre <cohomology groups> form a <local coefficient system> $\mathcal H^q(F;R)$ through the <fundamental group> action. For a <simply connected> base, it is constant. The stable terms are successive quotients of a finite filtration of each $H^n(E;R)$; reconstructing the <cohomology groups> can require solving <group extensions>. With commutative <ring> <coefficients>, the <cup product> makes the construction multiplicative.<The Serre Spectral Sequence by Allen Hatcher>[Theorem1.14 and §1.2]{ref}
For the <Hopf fibration> $S^1\to S^3\to S^2$ with <integer> <coefficients>, the only nonzero positions on $E_2$ are $(0,0),(0,1),(2,0),(2,1)$, each a copy of $\mathbb Z$. Since $H^1(S^3)=H^2(S^3)=0$, the map $d_2:E_2^{0,1}\to E_2^{2,0}$ must be an <isomorphism>. The survivors at $(0,0)$ and $(2,1)$ give the <cohomology> in degrees zero and three. This example shows how twisting changes the answer from the <cohomology> of a <Cartesian product>.
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