Significant Spectral Sequences

Why This Matters

Spectral sequences are the computational workhorses of algebraic topology—they transform impossibly complex calculations into systematic, page-by-page approximations. When you're faced with computing the homology of a fibration, the stable homotopy groups of spheres, or the cohomology of a sheaf, spectral sequences provide the machinery to break these problems into manageable pieces. You're being tested on your ability to recognize which spectral sequence applies to which situation, understand how filtrations give rise to successive approximations, and trace differentials through the pages until convergence.

Don't just memorize the names and $E_2$-pages—know what structural input each spectral sequence requires and what it computes. The key concepts here are fibrations and their associated long exact sequences, derived functors and their composition, filtrations on chain complexes, and the passage from algebra to topology. When an exam asks you to compute something, your first question should be: what's the underlying structure (fibration? group extension? sheaf?) and which spectral sequence exploits it?

---

Fibration-Based Spectral Sequences

These spectral sequences arise from fibrations $F \to E \to B$ and systematically relate the homology or cohomology of the total space to that of the base and fiber. The key insight is that a fibration provides a filtration of the total space by preimages of skeleta of the base.

Serre Spectral Sequence

• Computes $H_*(E)$ from $H_*(B)$ and $H_*(F)$—the $E_2$-page is $E_2^{p,q} = H_p(B; H_q(F))$, assuming $\pi_1(B)$ acts trivially on $H_*(F)$
• Differentials encode the twisting of the fibration—$d_r: E_r^{p,q} \to E_r^{p-r, q+r-1}$ captures how the fiber "varies" over the base
• The workhorse for computing homology of Lie groups and classifying spaces—essential for results like $H^*(K(\mathbb{Z}, n))$ and the cohomology of $BU(n)$

Eilenberg-Moore Spectral Sequence

• Handles pullbacks of fibrations—computes $H_*(E \times_B E')$ when you know the cohomology of $E$, $E'$, and $B$
• $E_2$-page involves $\text{Tor}$ over $H^*(B)$—specifically $E_2 = \text{Tor}_{H^*(B)}(H^*(E), H^*(E'))$
• Critical for loop space calculations—since $\Omega B \simeq * \times_B PB$, this spectral sequence computes $H_*(\Omega B)$ from $H^*(B)$

---

Stable Homotopy Spectral Sequences

These operate in the stable homotopy category, where spaces are replaced by spectra and ordinary homology gives way to generalized homology theories. The algebraic input involves Ext or Tor over structured ring spectra or their homotopy groups.

Adams Spectral Sequence

• Computes stable homotopy groups $\pi_*^s(X)$—the $E_2$-page is $E_2^{s,t} = \text{Ext}_{\mathcal{A}}^{s,t}(H^*(X; \mathbb{F}_p), \mathbb{F}_p)$ where $\mathcal{A}$ is the Steenrod algebra
• Convergence is to $p$-completed homotopy—you compute one prime at a time, then assemble via arithmetic
• The primary tool for $\pi_*^s(S^0)$—most of what we know about stable stems comes from Adams spectral sequence calculations

May Spectral Sequence

• Computes the $E_2$-page of the Adams spectral sequence—breaks the Ext calculation over $\mathcal{A}$ into smaller pieces using the filtration by powers of the augmentation ideal
• $E_1$-page involves Ext over associated graded—exploits the fact that $\text{gr}(\mathcal{A})$ is a primitively generated Hopf algebra
• Essential for practical Adams computations—without May, computing $\text{Ext}_{\mathcal{A}}$ directly would be intractable beyond low degrees

---

Generalized Cohomology Spectral Sequences

These spectral sequences connect ordinary cohomology to generalized cohomology theories like K-theory or cobordism. The filtration typically comes from the skeletal filtration of a CW complex.

Atiyah-Hirzebruch Spectral Sequence

• Computes $E^*(X)$ from $H^*(X)$ and $E^*(pt)$—for any generalized cohomology theory $E^*$, the $E_2$-page is $E_2^{p,q} = H^p(X; E^q(pt))$
• Differentials measure the failure of $E^*$ to be ordinary cohomology—for K-theory, the first nontrivial differential is related to Steenrod operations
• The bridge between singular cohomology and K-theory/cobordism—lets you leverage known $H^*(X)$ calculations to get at $K^*(X)$ or $MU^*(X)$

---

Sheaf-Theoretic Spectral Sequences

These arise from the derived functor perspective on sheaf cohomology and are fundamental in algebraic geometry. The key principle is that composing left-exact functors leads to a spectral sequence relating their derived functors.

Leray Spectral Sequence

• Computes $H^*(X; \mathcal{F})$ via a map $f: X \to Y$—the $E_2$-page is $E_2^{p,q} = H^p(Y; R^q f_* \mathcal{F})$ where $R^q f_*$ is the higher direct image
• Generalizes Serre to the sheaf setting—when $f$ is a fibration and $\mathcal{F}$ is constant, you recover Serre
• The tool for computing cohomology of fiber bundles in algebraic geometry—essential when the base has interesting sheaf cohomology

Grothendieck Spectral Sequence

• Relates derived functors of a composition $G \circ F$—if $F$ sends injectives to $G$-acyclics, then $E_2^{p,q} = R^p G(R^q F(-))$ converges to $R^{p+q}(G \circ F)(-)$
• Leray is a special case—take $F = f_*$ and $G = \Gamma(Y; -)$
• The abstract machine behind most spectral sequences in algebra—once you recognize a composition of functors, Grothendieck tells you there's a spectral sequence

---

Group Cohomology Spectral Sequences

These compute the cohomology of groups by exploiting exact sequences of groups. A short exact sequence $1 \to N \to G \to Q \to 1$ gives a "fibration" of classifying spaces $BN \to BG \to BQ$.

Hochschild-Serre Spectral Sequence

• Computes $H^*(G; M)$ from a normal subgroup $N \triangleleft G$—the $E_2$-page is $E_2^{p,q} = H^p(G/N; H^q(N; M))$
• Requires understanding the $G/N$-action on $H^*(N; M)$—this action encodes how conjugation in $G$ affects $N$-cohomology
• The group-theoretic analog of Serre—$BN \to BG \to B(G/N)$ is a fibration, and this is its Serre spectral sequence

Lyndon-Hochschild-Serre Spectral Sequence

• Same as Hochschild-Serre, different name—some authors distinguish by requiring $M$ to be a trivial module, but the terms are often used interchangeably
• Particularly useful for computing $H^*(G; \mathbb{Z})$—the case of trivial integer coefficients appears constantly in geometric group theory
• First step in computing cohomology of solvable groups—iterate over a composition series

---

Torsion and Coefficient Spectral Sequences

These spectral sequences arise from considering how cohomology changes with different coefficient groups. The Bockstein homomorphism $\beta: H^n(X; \mathbb{Z}/p) \to H^{n+1}(X; \mathbb{Z}/p)$ detects $p$-torsion in integral cohomology.

Bockstein Spectral Sequence

• Computes $H^*(X; \mathbb{Z})/\text{torsion} \otimes \mathbb{Z}/p$ from $H^*(X; \mathbb{Z}/p)$—the $E_1$-page is mod $p$ cohomology, and $d_1 = \beta$ is the Bockstein
• $E_\infty$ detects the "$p$-torsion-free part"—elements surviving to $E_\infty$ lift to integral classes
• Key for understanding the Universal Coefficient Theorem dynamically—the spectral sequence refines the UCT exact sequence

---

Quick Reference Table

---

Self-Check Questions

1. Both the Serre spectral sequence and the Hochschild-Serre spectral sequence have $E_2$-pages of the form "cohomology of the base with coefficients in cohomology of the fiber." What is the precise relationship between these two spectral sequences, and how does the classifying space functor mediate it?

2. You need to compute $K^*(X)$ for a CW complex $X$ whose ordinary cohomology $H^*(X; \mathbb{Z})$ you know. Which spectral sequence do you use, and what information beyond $H^*(X)$ do you need?

3. Compare the Grothendieck spectral sequence and the Leray spectral sequence. Which is more general? Under what hypotheses does Grothendieck specialize to Leray?

4. An FRQ asks you to compute $\text{Ext}_{\mathcal{A}}^{s,t}(\mathbb{F}_2, \mathbb{F}_2)$ for small $s$ and $t$. Which spectral sequence helps organize this computation, and what is its $E_1$-page?

5. You're given a group extension $1 \to \mathbb{Z}/p \to G \to \mathbb{Z}/p \to 1$ and asked to compute $H^*(G; \mathbb{F}_p)$. Which spectral sequence applies, what is the $E_2$-page, and what additional information determines the differentials?