For an open cover X = U ∪ V, a long exact sequence computes H_*(X) from H_*(U), H_*(V), H_*(U∩V). Iteration on a CW-cover provides a spectral-sequence approach; yields H_*(Sⁿ) by induction on n.
For an open cover X = U ∪ V, a long exact sequence computes H_*(X) from H_*(U), H_*(V), H_*(U∩V). Iteration on a CW-cover provides a spectral-sequence approach; yields H_*(Sⁿ) by induction on n.