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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 withThe 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.^[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.

For a Serre fibration $F\to E\to S^n$ with $n\ge 2$, the homological Serre spectral sequence has only columns $0,n$, and its only potentially nonzero differential isUsing an oriented fundamental class of $S^n$ identifies the source and target with $H_i(F)$ and $H_{i+n-1}(F)$. This gives the Wang homomorphism $\Delta$, and the resulting kernel/cokernel filtration pieces splice into the Wang sequence over a sphere.

For a fibration over the circle, the degree-preserving boundary map is instead the monodromy action minus the identity, $T_{\ast }-1:H_i(F)\to H_i(F)$, up to the chosen orientation sign. It arises from the cellular boundary with the fibre-homology local coefficient system. It is already incorporated in the $E_2$ page; there is no additional $d_1$ differential on an $E_2$-starting Serre spectral sequence. The distinction is visible for a product over $S^1$: its monodromy is the identity, so this boundary map is zero. The fundamental group action in the general construction is essential here.^[ref]