The Leray-Serre Spectral Sequence

The spectral sequence of a fibration is one of the most powerful computational tools in algebraic topology, systematically relating the homology of a fiber bundle to the homology of its base and fiber.

Statement

Theorem10.6Leray-Serre Spectral Sequence

Let $F \hookrightarrow E \xrightarrow{p} B$ be a Serre fibration with $B$ path-connected. Assume the fundamental group $\pi_1(B)$ acts trivially on $H_*(F; R)$ (e.g., $B$ is simply connected). Then there is a first-quadrant spectral sequence with $E^2_{p,q} = H_p(B; H_q(F; R)) \implies H_{p+q}(E; R)$ That is, the $E^2$ page consists of the homology of the base with coefficients in the homology of the fiber, and the spectral sequence converges to the homology of the total space.

The spectral sequence arises from filtering the total space $E$ by the preimages of the skeleta of $B$ (when $B$ is a CW complex). The differentials $d^r : E^r_{p,q} \to E^r_{p-r, q+r-1}$ encode increasingly subtle interactions between the base and fiber.

Cohomological Version

Theorem10.7Serre Spectral Sequence in Cohomology

Under the same hypotheses, there is a spectral sequence $E_2^{p,q} = H^p(B; H^q(F; R)) \implies H^{p+q}(E; R)$ Moreover, this spectral sequence is multiplicative: there is a product on each page compatible with the cup product on $H^*(E; R)$ .

ExampleCohomology of $\mathbb{CP}^n$

The Hopf fibration generalizes to $S^1 \to S^{2n+1} \to \mathbb{CP}^n$ . The Serre spectral sequence has $E_2^{p,q} = H^p(\mathbb{CP}^n) \otimes H^q(S^1)$ . Using the fact that $H^*(S^{2n+1})$ is known and analyzing the differentials, one deduces inductively that $H^*(\mathbb{CP}^n; \mathbb{Z}) \cong \mathbb{Z}[x]/(x^{n+1})$ with $|x| = 2$ .

Computing with the Spectral Sequence

ExampleLoop space of $S^n$

The path-loop fibration $\Omega S^n \to PS^n \to S^n$ has $PS^n$ contractible. The spectral sequence $H^p(S^n; H^q(\Omega S^n)) \implies H^{p+q}(PS^n) = 0$ (for $p+q > 0$ ) forces specific differentials to be isomorphisms, yielding: $H^*(\Omega S^n; \mathbb{Z}) \cong \begin{cases} \mathbb{Z}[x], \; |x| = n-1 & n \text{ even} \\ \mathbb{Z}[x]/(2x^2) \otimes \mathbb{Z}[y], \; |x| = n-1, |y| = 2n-2 & n \text{ odd} \end{cases}$ (with integer coefficients, the odd case is more subtle; over $\mathbb{Q}$ it simplifies).

RemarkSpectral sequences as successive approximations

A spectral sequence should be understood as a sequence of increasingly accurate approximations to the desired homology. The $E^2$ page is the first approximation (the "product" of base and fiber homology), and each subsequent differential $d^r$ encodes the obstruction to the fibration being a product at higher and higher levels of precision.