Wang sequence for a fibration over the circle

Fix $M_q=H_q(F;R)$. The cellular local chain complex of the oriented circle with coefficients in the transport system has one copy of $M_q$ in degrees one and zero. With the convention that the positive edge has initial incidence $+1$ and terminal incidence $-1$, [F2] makes its boundary $1-T_q$. Therefore $E_{0,q}^2=\mathrm{coker}(1-T_q),E_{1,q}^2=\ker (1-T_q),$ and $E_{a,q}^2=0$ for $a\mathrm{\notin }\{0,1\}$. Reversing the edge interchanges its endpoint incidences and changes the differential to $T_q-1$.

Every Serre differential from page two onward changes the first coordinate by at least two, so the two-column support gives a zero source or target. Hence $E^2=E^{\mathrm{\infty }}$. In total degree $q$, the finite filtration of [F1] and its edges in [F3] give the natural short exact sequence $0⟶\mathrm{coker}(1-T_q)⟶H_q(E;R)⟶\ker (1-T_{q-1})⟶0.\text{(2)}$ The first map in (2) is the fiber-axis inclusion after quotienting by $1-T_q$; the second is the base-column quotient followed by the inclusion of the kernel.

Compose the quotient $M_q\to \mathrm{coker}(1-T_q)$ with the first arrow of (2), and compose the second arrow of (2) with $\ker (1-T_{q-1})\hookrightarrow M_{q-1}$. The kernel and image definitions now give, in order, $\ker (M_q\to H_q(E))=\mathrm{im}(1-T_q),$ $\mathrm{im}(M_q\to H_q(E))=\ker (H_q(E)\to M_{q-1}),$ and $\mathrm{im}(H_q(E)\to M_{q-1})=\ker (1-T_{q-1})\mathrm{.}$ Joining these identities for all $q$ proves exactness of (1).

A map of fibrations over the oriented circle gives a morphism of local systems, so its fiber map commutes with every $T_q$. Naturality in [F1]–[F4] makes the quotient, kernel, and filtration arrows in (2) commute; therefore it gives a morphism of the long exact sequences. The same proof with the reverse cellular generator gives $T_q-1$, and multiplication by $-1$ identifies the two versions.

If the fiber or total space is empty, all stalks and groups are zero. The zero ring and the zero module give the zero exact sequence. For $q=0$, the right-hand term $H_{-1}(F;R)$ is zero; negative degrees continue by zeros. If $T_q=1$, the two endomorphisms are zero and (2) is still the asserted kernel-cokernel extension. A one-element or zero homology group causes no exception. Degenerate cellular or singular representatives contribute zero in the normalized page class. Both cell endpoints, both orientation signs, both columns, both ends of (2), and all three exactness positions were checked. Every construction uses finite kernels, cokernels, and induced maps, so no AC is used. There is no iff assertion and no splitting of (2) is claimed.