Wang exact sequence Let F −→ E −→ Sn be a fiber bundle with n ≥ 2 and F path connected. Then there is a long exact sequence
· · · −→ Hk(E) −→ Hk(F ) −→ Hk−n+1(F ) −→ Hk+1(E) −→ · · ·
called the Wang exact sequence. This sequence is derived from the spectral
sequence for the fiber bundle, which in this case has only one non-trivial differential. There is an analogous sequence for homology. One can use the Wang sequence to compute the homology of the based loop space of a sphere.