 Wang homomorphism

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]