Wang sequence

Remarks

The base $S^n$ is simply connected because $n \ge 2$, so the Serre spectral sequence has $E^2_{s,t} = H_s(S^n; H_t(F))$. This is nonzero only for $s = 0$ and $s = n$, so the only possible differential is $d^n \colon E^n_{n,t} \to E^n_{0,t+n-1}$. Unfolding this two-column spectral sequence gives the long exact sequence, and the map $H_{k-n}(F) \to H_{k-1}(F)$ is this differential. A dual sequence holds in cohomology.

Example: the path space fibration $\Omega S^n \to P S^n \to S^n$ has contractible total space, where $\Omega S^n$ is the loop space. Exactness then gives isomorphisms $H_{k-n}(\Omega S^n) \cong H_{k-1}(\Omega S^n)$ for $k \ge 2$. Since $H_0(\Omega S^n) = \Z$, one gets $H_j(\Omega S^n) \cong \Z$ when $j$ is a multiple of $n - 1$ and $H_j(\Omega S^n) = 0$ otherwise.

Going deeper

For a student

The Wang sequence is the fibration analogue of the Mayer–Vietoris sequence, obtained by cutting the sphere into two discs. It is the base-sphere counterpart of the Gysin sequence, which treats fibrations whose fibre is a sphere. Its classical use is the computation of the homology of loop spaces of spheres, as in the example above.