8.4 The Serre spectral sequence

The Serre spectral sequence is a powerful tool in algebraic topology that links the homology of a fibration's base space, fiber, and total space. It's like a mathematical recipe that helps us understand how these spaces fit together.

This spectral sequence starts with the homology of the base and fiber, then uses a series of steps to calculate the total space's homology. It's super useful for figuring out complex spaces by breaking them down into simpler parts.

The Serre spectral sequence

Definition and main properties

• The Serre spectral sequence is a powerful tool in algebraic topology that establishes a relationship between the homology of the base space, fiber, and total space of a fibration
• Given a fibration $F \to E \to B$, the Serre spectral sequence consists of a sequence of pages $(E_r)$, each containing a two-dimensional array of abelian groups $E_{p,q}^r$, where $p$ and $q$ are the indices of the array
• The pages are connected by differentials $d_r: E_{p,q}^r \to E_{p+r,q-r+1}^r$, satisfying $d_r \circ d_r = 0$, which gives rise to the next page $E_{r+1}$ as the homology of $E_r$ with respect to $d_r$
• The Serre spectral sequence converges to the homology of the total space $E$, with the limit term $E_\infty$ being the associated graded of a filtration on $H_*(E)$

Structure of the $E_2$ page

• The $E_2$ page of the Serre spectral sequence has terms $E_{p,q}^2 \cong H_p(B; H_q(F))$, where $H_q(F)$ is the local coefficient system on $B$ obtained from the homology of the fiber $F$
  • This means that the $E_2$ page encodes the homology of the base space with coefficients in the homology of the fiber
  • The local coefficient system captures the twisting or variation of the homology of the fiber as it moves along the base space
• The $E_2$ page provides a starting point for computing the homology of the total space by successively applying the differentials and taking homology

Applications of the Serre spectral sequence

Computing homology and cohomology of fibrations

• To compute the homology or cohomology of the total space $E$ using the Serre spectral sequence:
  1. Identify the $E_2$ page using the homology of the base $B$ and the homology of the fiber $F$
  2. Determine the differentials on each page of the spectral sequence using the structure of the fibration and the convergence properties
  3. Compute the homology or cohomology groups on each page by taking the kernel of the outgoing differential modulo the image of the incoming differential
  4. Continue this process until the spectral sequence collapses (i.e., when all differentials are zero) and the terms stabilize
• The limit term $E_\infty$ gives the associated graded of the homology or cohomology of the total space $E$, which can be used to reconstruct the actual homology or cohomology groups
  • In some cases, the spectral sequence may collapse at an early stage, allowing for a complete computation of the homology or cohomology of $E$

Examples and applications

• The Serre spectral sequence can be applied to compute the homology and cohomology of various fibrations, such as:
  • The Hopf fibration $S^1 \to S^3 \to S^2$, which is a principal $S^1$-bundle over $S^2$
  • The path space fibration $\Omega X \to PX \to X$, where $PX$ is the space of paths in $X$ and $\Omega X$ is the loop space of $X$
  • The Borel construction $EG \times_G F \to BG$ for a $G$-space $F$, where $EG$ is a contractible space with a free $G$-action and $BG$ is the classifying space of $G$
• The Serre spectral sequence can also be used to derive important results in algebraic topology, such as the cohomology of Eilenberg-MacLane spaces and the Hurewicz theorem relating homotopy groups to homology groups

Structure and convergence of the Serre spectral sequence

Relationship between base, fiber, and total space

• The structure of the Serre spectral sequence reflects the intricate relationship between the homology of the base, fiber, and total space of a fibration
• The $E_2$ page captures the homology of the base with coefficients in the homology of the fiber, incorporating the twisting or local coefficient system induced by the fibration
• The differentials in the spectral sequence arise from the long exact sequence of the fibration and the multiplicative structure of the spectral sequence, encoding the interactions between the homology of the base and the fiber

Convergence and the associated graded

• Convergence of the Serre spectral sequence means that the limit term $E_\infty$ is the associated graded of a filtration on the homology of the total space $E$
  • The filtration on $H_*(E)$ is induced by the skeletal filtration of the base space $B$, with the associated graded pieces corresponding to the homology of the fibers over each skeleton
  • This filtration provides a decomposition of the homology of the total space in terms of the contributions from the base and the fiber
• The edge homomorphisms of the spectral sequence, which are maps between the homology of the base, the homology of the total space, and the homology of the fiber, provide additional information about their relationships and can be used to extract specific homology groups or maps between them

Serre spectral sequence vs other spectral sequences

Comparison with other spectral sequences

• The Serre spectral sequence is one of several spectral sequences used in algebraic topology to compute homology and cohomology of spaces related by certain constructions, such as fibrations, cofibrations, or group actions
• The Atiyah-Hirzebruch spectral sequence relates the ordinary homology of a space to its generalized homology theories (K-theory, cobordism)
  • Its $E_2$ term is given by $E_{p,q}^2 \cong H_p(X; h_q(pt))$, where $h_*(pt)$ is the coefficient ring of the generalized homology theory
• The Eilenberg-Moore spectral sequence converges to the homology of the total space of a fibration, with the $E_2$ term given by the tor groups of the homology of the base and the homology of the fiber

Generalizations and variants

• The Leray-Serre spectral sequence generalizes the Serre spectral sequence to fibrations with non-simply connected base spaces
  • It involves local coefficient systems and the action of the fundamental group of the base on the homology of the fiber
• Variants of the Serre spectral sequence exist for cohomology, with the $E_2$ page given by $E_2^{p,q} \cong H^p(B; H^q(F))$, and for fibrations of spectra, which are used in stable homotopy theory
• Comparing the different spectral sequences provides insights into the relationships between various homology and cohomology theories and helps in choosing the most effective tool for computing the invariants of a given space or map