= Wang homomorphism
{c}
{title2=$\Delta:H_i(F)\to H_{i+n-1}(F)$}
For a <Serre fibration> $F\to E\to S^n$ with $n\geq2$, the homological <Serre spectral sequence> has only columns $0,n$, and its only potentially nonzero differential is
$$
d_n:E^n_{n,i}\longrightarrow E^n_{0,i+n-1}.
$$
Using 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, $T_*-1:H_i(F)\to H_i(F)$, 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.<The Serre Spectral Sequence by Allen Hatcher>[§1.1, fibre transport and local coefficients]{ref}
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