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Current page: E₁
Active differentials: —
Surviving classes: —
Example: Serre SS for S¹→ES¹→BS¹
Fibration S¹ → * → ℂP∞
E₂^{p,q} = H^p(ℂP∞;H^q(S¹))
= ℤ for (p,q)=(2k,0),(2k,1)
Differentials d₂: E₂^{0,1}→E₂^{2,0}
All isomorphisms → H*(*)=ℤ in deg 0
Spectral sequence {Eᵣ,dᵣ}: bigraded modules with dᵣ: Eᵣ^{p,q} → Eᵣ^{p+r,q-r+1}, dᵣ²=0.
Eᵣ₊₁ = H(Eᵣ,dᵣ) = ker(dᵣ)/im(dᵣ).
Convergence: Eᵣ stabilizes to E∞, which gives the associated graded of a filtration on the limit H*.
Serre SS: fibration F→E→B gives E₂^{p,q}=H^p(B;H^q(F))⟹H^{p+q}(E).
AHSS, Adams SS: compute stable homotopy groups.