A spectral sequence has E²_{2,0} = ℤ/2ℤ and E²_{0,2} = ℤ/2ℤ, with all differentials d_r = 0 for r ≥ 2, so E∞ = E². Both groups contribute to H₂ of the total complex via a filtration. What is H₂?
Aℤ/2ℤ ⊕ ℤ/2ℤ, because we read off H₂ directly as the direct sum of the E∞ entries
BEither ℤ/2ℤ ⊕ ℤ/2ℤ or ℤ/4ℤ, depending on how the extension 0 → ℤ/2ℤ → H₂ → ℤ/2ℤ → 0 resolves
Cℤ/4ℤ, because the two ℤ/2ℤ factors combine into the next cyclic group
DUndetermined — the spectral sequence gives insufficient information even in principle
The E∞ page gives the *associated graded* of the filtered homology, not the homology directly. Knowing E∞_{2,0} = ℤ/2ℤ and E∞_{0,2} = ℤ/2ℤ tells you there is a short exact sequence 0 → ℤ/2ℤ → H₂ → ℤ/2ℤ → 0, but this does not uniquely determine H₂. There are exactly two non-isomorphic extensions of ℤ/2ℤ by ℤ/2ℤ: the direct sum ℤ/2ℤ ⊕ ℤ/2ℤ (split extension) and ℤ/4ℤ (non-split). Determining which requires extra-spectral information. This is the extension problem — the essential difficulty that survives even after the spectral sequence fully converges.
Question 2 Multiple Choice
What does the differential d_r on the E^r page of a spectral sequence do?
AIt computes H_*(C•) directly by differentiating the filtered chain complex
BIt maps E^r_{p,q} → E^r_{p−r, q+r−1}, and taking its homology produces the next page E^{r+1}
CIt identifies which elements have already survived from the E¹ page to E^r
DIt assembles the E∞ entries back into the total homology by solving the extension problems
Each page E^r has a differential d_r that goes r steps to the left and r−1 steps up in the bigraded diagram: d_r: E^r_{p,q} → E^r_{p−r, q+r−1}. This differential satisfies d_r² = 0, so it is a chain map on the bigraded group. The homology of (E^r, d_r) gives the next page E^{r+1}: elements that are cycles for d_r (kernel) modulo boundaries (image) survive to the next page. As r increases, differentials reach further across the diagram, killing more elements. The process continues until all differentials are zero — at which point the sequence has stabilized at E∞.
Question 3 True / False
The E∞ page of a convergent spectral sequence gives the associated graded of the filtered homology, so knowing E∞ may still leave the actual homology group undetermined due to extension problems.
TTrue
FFalse
Answer: True
Convergence means E∞_{p,q} ≅ F_p H_{p+q}/F_{p-1} H_{p+q} — the graded pieces of the filtration on homology. Knowing the graded pieces tells you the homology up to extensions. For free abelian groups (e.g., ℤ), all extensions split and the homology is the direct sum of the E∞ entries. But for groups with torsion, non-trivial extensions can exist: two copies of ℤ/2ℤ in E∞ might assemble into ℤ/2ℤ ⊕ ℤ/2ℤ or ℤ/4ℤ. Resolving extension problems often requires going back to the original complex or using additional algebraic tools.
Question 4 True / False
If a spectral sequence collapses at the E² page (most differentials d_r = 0 for r ≥ 2), then the target homology is uniquely determined as the direct sum of most E² entries in total degree n.
TTrue
FFalse
Answer: False
Collapse at E² means E∞ = E², which gives you the associated graded of the filtered homology — but extension problems between those graded pieces may still prevent uniquely determining the homology group. The direct sum is one possibility (the split extension), but non-trivial extensions may exist. For example, if E²_{2,0} = ℤ/2ℤ and E²_{0,2} = ℤ/2ℤ both contribute to H₂, the spectral sequence collapsing at E² only tells you H₂ fits into 0 → ℤ/2ℤ → H₂ → ℤ/2ℤ → 0 — not which extension it is. Unique determination from E∞ is guaranteed only when all relevant extension groups Ext¹ vanish (e.g., when all E∞ entries are free).
Question 5 Short Answer
Explain what it means for a spectral sequence to 'collapse' at the E² page, and why collapsing does not guarantee that the homology is fully determined.
Think about your answer, then reveal below.
Model answer: Collapsing at E² means all differentials d_r = 0 for every r ≥ 2, so no further elements are killed after the E² page — E∞ = E². This simplifies computation enormously: you don't need to track which elements survive higher differentials. However, the spectral sequence only produces the associated graded of the filtered homology, not the homology itself. Even with E∞ known, the homology sits in short exact sequences 0 → F_{p-1}H_n → F_pH_n → E∞_{p,n-p} → 0 that may not split. Different extensions give non-isomorphic groups with the same associated graded. Determining the actual extension class requires information not contained in the spectral sequence alone.
The Leray-Serre spectral sequence for a product fibration collapses at E² and gives the Künneth formula — an example where extensions happen to split. For non-trivial fibrations, the spectral sequence may also collapse but leave genuine ambiguity. Recognizing when extensions split (e.g., when all groups involved are free abelian, or when the sequence lives over a field) is a key skill in applying spectral sequences.