Topology
Spectral Sequences: Introduction
Consider a machine that iteratively "refines" its knowledge of a space, page by page, until it computes the exact answer. That is the spectral sequence: the universal computational tool of algebraic topology.
- **Computing homotopy groups of spheres:** Adams used the Adams spectral sequence to systematically compute π*(S), stable homotopy groups of spheres
- **Algebraic K-theory:** The Bloch-Lichtenbaum spectral sequence connects motivic cohomology to algebraic K-theory
- **Deformation theory:** In algebraic geometry, spectral sequences count deformations of complex manifolds
Предварительные знания
Why Spectral Sequences?
The Serre spectral sequence (1951) computed homotopy groups used in string theory: Adams found all stable πₙˢ for n ≤ 13. A **spectral sequence** is a machine for computing the homology (or cohomology) of a complex space from the homology of its "pieces". Given a filtration of a complex or a fiber bundle, a spectral sequence automatically organizes the computation.
Motivation: given a fiber bundle F → E → B, how are H*(E), H*(F), and H*(B) related? The Mayer-Vietoris exact sequence does not apply here. The Serre spectral sequence gives a complete answer.
Spectral sequences were invented by Jean Leray in the 1940s while he was a German prisoner of war. He developed sheaf theory and spectral sequences, hiding from guards that he was doing "pure" (rather than war-applicable) mathematics.
What is the Serre spectral sequence of a fibration F → E → B used for?
Pages and Differentials
A spectral sequence is a sequence of "pages" E_r, r = 0, 1, 2, ..., where each page is a bigraded abelian group equipped with a differential d_r.
Key fact: if all differentials d_r are trivial starting from r=2 (the sequence **degenerates** at E₂), then E₂ = E_∞ and the computation simplifies considerably.
What is E_{r+1}^{p,q} in a spectral sequence?
Convergence and E_∞
A spectral sequence **converges** to H*(E) if E_∞ is the page at which all differentials vanish. Then there is a filtration on H_n(E), and the associated graded pieces equal E_∞^{p,q} for p+q=n.
If the coefficients are in a field (Q, Fp) and the bundle has "good" properties (e.g., H*(B) or H*(F) concentrated in even degrees), then often E₂ = E_∞. For bundles over simply connected spaces this frequently happens automatically.
Why does knowing E_∞ not always let us recover H*(E) uniquely?
Example: the Hopf Fibration
Let us apply the Serre spectral sequence to the Hopf fibration S¹ → S³ → S². This will show concretely how the machine works.
The nontrivial d₂ in the Hopf fibration reflects the nontriviality of the fibration itself! If the bundle were trivial (S¹ × S²), all differentials would be zero and we would get H*(S¹ × S²) = H*(S¹) ⊗ H*(S²), which is far from H*(S³).
In the Serre spectral sequence of the Hopf fibration S¹ → S³ → S², the differential d₂: E₂^{2,0} → E₂^{0,1} is an isomorphism. What does this mean topologically?
Key Ideas
- **Spectral sequence**, a sequence of pages E_r with differentials; E_{r+1} = H(E_r, d_r)
- **E₂^{p,q} = Hₚ(B; Hq(F))**, the initial page for fibrations (Serre SS)
- **Degeneration at E₂**, if all d_r = 0 for r ≥ 2, then E₂ = E_∞
- **Extension problem**, E_∞ determines H*(E) only up to group extensions
Related Topics
Spectral sequences are the computational tool for many theories:
- Fiber Bundles — The Serre spectral sequence is the main tool for fiber bundles
- Homology — Spectral sequences compute homology of complex spaces from simpler pieces
- K-theory — The Atiyah-Hirzebruch spectral sequence connects K-theory to ordinary cohomology
Вопросы для размышления
- Why does a spectral sequence not give H*(E) directly, but only up to extensions?
- How does a nontrivial differential d₂ reflect the nontriviality of the bundle?
- What does it mean for a spectral sequence to "degenerate" at the second page?