Spectral Sequences - Applications

Spectral sequences are indispensable tools in algebraic topology, algebraic geometry, and beyond.

Theorem7.14Computing Homotopy Groups via Serre

The Serre spectral sequence for the path-loop fibration $\Omega X \to PX \to X$ : $E_2^{p,q} = H^p(X; \pi_q(\Omega X)) \Rightarrow H^{p+q}(PX) = 0$

allows computation of homotopy groups from cohomology. Since $PX$ is contractible, edge maps give information about $\pi_*(X)$ .

ExampleHurewicz Theorem via Spectral Sequences

The Serre spectral sequence provides an elegant proof of the Hurewicz theorem: for a simply connected space $X$ , the Hurewicz map $\pi_n(X) \to H_n(X)$ is an isomorphism for $n \leq k$ if and only if $\pi_i(X) = 0$ for $i < k$ .

Theorem7.15Atiyah-Hirzebruch Spectral Sequence

For a generalized cohomology theory $h^*$ and CW complex $X$ : $E_2^{p,q} = H^p(X; h^q(\text{pt})) \Rightarrow h^{p+q}(X)$

This computes generalized cohomology from singular cohomology with coefficients in $h^*(\text{pt})$ .

Remark

The Atiyah-Hirzebruch spectral sequence is fundamental in K-theory, cobordism theory, and other generalized cohomology theories, reducing computations to ordinary cohomology.

Theorem7.16Adams Spectral Sequence

For computing stable homotopy groups of spheres: $E_2^{s,t} = \text{Ext}_{\mathcal{A}}^{s,t}(\mathbb{F}_p, \mathbb{F}_p) \Rightarrow \pi_{t-s}(S^0)_{(p)}$

where $\mathcal{A}$ is the Steenrod algebra. This is one of the most powerful tools in stable homotopy theory.

ExampleCohomology of Lie Algebras

For a Lie algebra $\mathfrak{g}$ with ideal $\mathfrak{h}$ , the Hochschild-Serre spectral sequence: $E_2^{p,q} = H^p(\mathfrak{g}/\mathfrak{h}, H^q(\mathfrak{h}, M)) \Rightarrow H^{p+q}(\mathfrak{g}, M)$

computes Lie algebra cohomology via quotients.

Theorem7.17Hodge-to-de Rham Spectral Sequence

For a smooth scheme $X$ over a field of characteristic $p > 0$ : $E_1^{p,q} = H^q(X, \Omega^p_{X/k}) \Rightarrow H^{p+q}_{\text{dR}}(X/k)$

This relates algebraic de Rham cohomology to Hodge cohomology, fundamental in arithmetic geometry.