Spectral Sequences

Jean-Pierre Serre in 1951 used spectral sequences to compute infinitely many nonzero homotopy groups of spheres -- overturning the intuition of the era. A spectral sequence is a machine for computing homology through iterated approximations.

• Algebraic topology: computing H*(E) for fibrations, foundation of characteristic class theory
• Algebraic geometry: the Leray spectral sequence computes cohomology of sheaf pushforwards
• Stable homotopy theory: the Adams spectral sequence is the main computational tool

Предварительные знания

• Previous lesson

Spectral Sequences: Definition

Jean-Pierre Serre in 1951 computed H*(K(Z,n)) using spectral sequences, producing infinitely many nonzero groups π_n(S²) -- a result once thought impossible. A spectral sequence is a sequence of pages (E_r, d_r), r ≥ 2, where each page is a bicomplex with differential d_r: E_r^{p,q} → E_r^{p+r,q-r+1}, and E_{r+1} = H(E_r, d_r).

Convergence and the Comparison Theorem

Adams in 1958 used the Adams spectral sequence to prove that the Hopf invariant 1 problem has solutions only in dimensions 1, 2, 4, 8 -- meaning normed division algebras over R exist only in those dimensions. A spectral sequence converges when the filtration is finite, or when for some r all differentials d_r = 0 and E_{r+1} = E_r =: E_∞.

Key Ideas

• Spectral sequence: pages (E_r, d_r), d_r: E_r^{p,q} -> E_r^{p+r,q-r+1}
• E_inf = associated graded of the filtration on H*(C)
• Serre spectral sequence: E_2 = H^p(B; H^q(F)) => H*(E)
• Comparison theorem: isomorphism on E_2 implies isomorphism on all pages
• chi(E) = chi(F)*chi(B) -- Euler characteristic is multiplicative for fibrations

Further Directions

The concepts studied here open paths to deeper areas of mathematics.

• aa-28-alg-topology — extends

Вопросы для размышления

• Work through a concrete computation using the methods from this lesson.
• How do the ideas here connect to other areas of mathematics covered earlier?