8.4 Applications in homological algebra

Spectral sequences are powerful tools in homological algebra, connecting different cohomology theories. They allow us to compute complex structures by breaking them down into simpler pieces and using successive approximations.

From Leray and Serre to Eilenberg-Moore and Adams, these sequences tackle various algebraic and topological problems. They're essential for understanding group cohomology, fibrations, and even stable homotopy groups of spheres.

Spectral Sequences in Homological Algebra

Key Spectral Sequences for Homology and Cohomology

• Leray spectral sequence relates the cohomology of a space with the cohomology of a sheaf on that space
  • Enables computing cohomology groups using a fibration $F \to E \to B$ by successive approximations
  • Starts with $E_2^{p,q} = H^p(B; H^q(F))$ and converges to $H^{p+q}(E)$
• Serre spectral sequence is a spectral sequence that relates the homology of the base space and the homology of the fiber to the homology of the total space in a fibration
  • For a fibration $F \to E \to B$, it has $E^2_{p,q} = H_p(B; H_q(F))$ and converges to $H_{p+q}(E)$
  • Powerful tool for computing homology groups of a space when it's realized as a fibration (Hopf fibration of spheres)
• Atiyah-Hirzebruch spectral sequence relates a generalized cohomology theory of a space to its ordinary cohomology
  • For a spectrum $h$ and CW complex $X$, it starts with $E_2^{p,q} = H^p(X; \pi_q(h))$ and converges to $h^{p+q}(X)$
  • Allows computing extraordinary cohomology groups like K-theory or cobordism from ordinary cohomology (stable homotopy groups of spheres)

Spectral Sequences for Ring and Module Structures

• Eilenberg-Moore spectral sequence relates the cohomology of a fiber space to the cohomology of the base and total space for a fibration of spaces with additional structure
  • Applies when cohomology of fiber, total space and base have compatible ring or module structures
  • Starts with $E_2^{p,q} = \text{Tor}^{H^*(B)}_{p,q}(H^*(E), k)$ and converges to $H^{p+q}(F)$ as modules over $H^*(B)$
• Adams spectral sequence computes stable homotopy groups of spheres from Ext groups in the category of modules over the Steenrod algebra
  • $E_2^{s,t} = \text{Ext}_{\mathcal{A}}^{s,t}(\mathbb{F}_p, \mathbb{F}_p) \Rightarrow \pi_{t-s}^S(\mathbb{S}^0)_{(p)}$
  • Connects stable homotopy theory to homological algebra of the Steenrod algebra (graded Hopf algebra encoding cohomology operations)

Group Cohomology Spectral Sequences

Spectral Sequences for Group Extensions

• Hochschild-Serre spectral sequence relates the cohomology of a group extension to the cohomology of the subgroup and quotient group
  • For a group extension $1 \to H \to G \to K \to 1$ it has $E_2^{p,q} = H^p(K; H^q(H; M))$ converging to $H^{p+q}(G; M)$
  • Allows computing group cohomology using the cohomology of smaller pieces $H$ and $K$ (when $G$ is a semidirect product)
• Lyndon-Hochschild-Serre spectral sequence is a specific case of the Hochschild-Serre spectral sequence
  • Applies to group cohomology with coefficients in a trivial module
  • Has $E_2^{p,q} = H^p(K; H^q(H; \mathbb{Z}))$ and converges to $H^{p+q}(G; \mathbb{Z})$
  • Useful for analyzing the cohomological dimension of solvable groups (successive extensions by abelian groups)