Editing Spectral sequence (section)

=== Wang sequence ===
The computation in the previous section generalizes in a straightforward way. Consider a [[fibration]] over a sphere:
:<math>F \overset{i}\to E \overset{p}\to S^n</math>
with ''n'' at least 2. There is the [[Serre spectral sequence]]:
:<math>E^2_{p, q} = H_p(S^n; H_q(F)) \Rightarrow H_{p+q}(E)</math>;
that is to say, <math>E^{\infty}_{p, q} = F_p H_{p+q}(E)/F_{p-1} H_{p+q}(E)</math> with some filtration <math>F_\bullet</math>.

Since <math>H_p(S^n)</math> is nonzero only when ''p'' is zero or ''n'' and equal to '''Z''' in that case, we see <math>E^2_{p, q}</math> consists of only two lines <math>p = 0,n</math>, hence the <math>E^2</math>-page is given by
:<math>\begin{matrix}
& \vdots & \vdots & \vdots &  & \vdots  & \vdots & \vdots & \\
\cdots & 0 & E^2_{0,2} & 0 & \cdots & 0 & E^2_{n,2} & 0 & \cdots \\
\cdots & 0 & E^2_{0,1} & 0 & \cdots & 0  & E^2_{n,1} & 0 & \cdots \\
\cdots & 0 & E^2_{0,0} & 0 & \cdots & 0  & E^2_{n,0} & 0 & \cdots \\
\end{matrix}</math>
Moreover, since
:<math>E^2_{p, q} = H_p(S^n;H_q(F)) = H_q(F)</math>
for <math>p = 0,n</math> by the [[universal coefficient theorem]], the <math>E^2</math> page looks like
:<math>\begin{matrix}
& \vdots & \vdots & \vdots &  & \vdots  & \vdots & \vdots & \\
\cdots & 0 & H_2(F) & 0 & \cdots & 0 & H_2(F) & 0 & \cdots \\
\cdots & 0 & H_1(F) & 0 & \cdots & 0  & H_1(F) & 0 & \cdots \\
\cdots & 0 & H_0(F) & 0 & \cdots & 0  & H_0(F) & 0 & \cdots \\
\end{matrix}</math>
Since the only non-zero differentials are on the <math>E^n</math>-page, given by
:<math>d^n_{n,q}:E^n_{n,q} \to E^n_{0,q+n-1}</math>
which is
:<math>d^n_{n,q}:H_q(F) \to H_{q+n-1}(F)</math>
the spectral sequence converges on <math>E^{n+1} = E^{\infty}</math>. By computing <math>E^{n+1}</math> we get an exact sequence
:<math>0 \to E^{\infty}_{n, q-n} \to E^n_{n, q-n} \overset{d}\to E^n_{0, q-1} \to E^{\infty}_{0, q-1} \to 0.</math>
and written out using the homology groups, this is
:<math>0 \to E^{\infty}_{n, q-n} \to H_{q-n}(F) \overset{d}\to H_{q-1}(F) \to E^{\infty}_{0, q-1} \to 0.</math>
To establish what the two <math>E^\infty</math>-terms are, write <math>H = H(E)</math>, and since <math>F_1 H_q/F_0 H_q = E^{\infty}_{1, q - 1} = 0</math>, etc., we have: <math>E^{\infty}_{n, q-n} = F_n H_q / F_0 H_q</math> and thus, since <math>F_n H_q = H_q</math>,
:<math>0 \to E^{\infty}_{0, q} \to H_q \to E^{\infty}_{n, q - n} \to 0.</math>
This is the exact sequence
:<math>0 \to H_q(F) \to H_q(E) \to H_{q-n}(F)\to 0.</math>
Putting all calculations together, one gets:<ref>{{harvnb|Weibel|1994|loc=Application 5.3.5.}}</ref>
:<math>\dots \to H_q(F) \overset{i_*}\to H_q(E) \to H_{q-n}(F) \overset{d}\to H_{q-1}(F) \overset{i_*}\to H_{q-1}(E) \to H_{q-n -1}(F) \to \dots</math>
(The [[Gysin sequence]] is obtained in a similar way.)