8.2 Serre spectral sequence

The Serre spectral sequence is a powerful tool in algebraic topology that connects the cohomology of fibrations. It breaks down complex spaces into simpler components, allowing us to compute cohomology by analyzing relationships between base spaces, fibers, and total spaces.

This sequence is particularly useful for fiber bundles, Eilenberg-MacLane spaces, and classifying spaces. It provides a systematic approach to understanding cohomological structures, though challenges like computational complexity and determining differentials can arise in practice.

Definition of Serre spectral sequence

• The Serre spectral sequence is a powerful tool in algebraic topology that relates the cohomology of the base space, fiber, and total space of a fibration
• It provides a systematic way to compute the cohomology of a space by breaking it down into simpler pieces and analyzing the relationships between them
• The spectral sequence consists of a series of pages, each containing cohomology groups, connected by differentials that encode information about the fibration structure

Serre spectral sequence for fibrations

Leray-Serre spectral sequence

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• The Leray-Serre spectral sequence is a specific instance of the Serre spectral sequence that applies to fibrations
• It relates the cohomology of the base space $B$, the cohomology of the fiber $F$, and the cohomology of the total space $E$ in a fibration $F \to E \to B$
• The $E_2$ page of the spectral sequence has terms $E_2^{p,q} = H^p(B; H^q(F))$, which involve the cohomology of the base space with coefficients in the cohomology of the fiber
  • The differentials on the $E_2$ page are determined by the twisting of the fibration and provide information about the cohomology of the total space

Serre spectral sequence of fiber bundles

• Fiber bundles are a special case of fibrations where the total space is locally trivial, meaning it locally looks like a product of the base space and the fiber
• The Serre spectral sequence simplifies for fiber bundles, as the cohomology of the fiber is constant over the base space
• In this case, the $E_2$ page takes the form $E_2^{p,q} = H^p(B) \otimes H^q(F)$, a tensor product of the cohomology of the base and fiber
  • The differentials provide information about the global twisting of the bundle and how it affects the cohomology of the total space

Convergence of Serre spectral sequence

Conditions for convergence

• The Serre spectral sequence is said to converge if there exists an $r$ such that the differentials $d_r, d_{r+1}, \ldots$ are all zero
• Convergence implies that the spectral sequence stabilizes at the $E_r$ page, and the cohomology of the total space can be determined from this stable page
• Sufficient conditions for convergence include the base space being simply connected or the fiber being of finite type (having finitely generated cohomology in each degree)

Consequences of convergence

• When the Serre spectral sequence converges, it provides a filtration of the cohomology of the total space
• The associated graded of this filtration is isomorphic to the stable page of the spectral sequence
• Convergence allows for the computation of the cohomology of the total space in terms of the cohomology of the base and fiber, along with additional information encoded in the differentials

Applications of Serre spectral sequence

Computing cohomology of fiber bundles

• The Serre spectral sequence is particularly useful for computing the cohomology of fiber bundles, such as vector bundles, principal bundles, and associated bundles
• By understanding the cohomology of the base space and the fiber, along with the twisting of the bundle, the spectral sequence provides a systematic approach to determine the cohomology of the total space
• Examples include computing the cohomology of complex projective spaces $\mathbb{CP}^n$ (which are fiber bundles over $\mathbb{CP}^{n-1}$ with fiber $S^2$) and Stiefel manifolds (which are fiber bundles over Grassmann manifolds)

Cohomology of Eilenberg-MacLane spaces

• Eilenberg-MacLane spaces $K(G, n)$ are spaces with a single non-trivial homotopy group $\pi_n(K(G, n)) = G$ and contractible higher homotopy groups
• The cohomology of Eilenberg-MacLane spaces can be computed using the Serre spectral sequence by considering the path-loop fibration $\Omega K(G, n) \to PK(G, n) \to K(G, n)$
• The spectral sequence reveals that the cohomology of $K(G, n)$ is isomorphic to the group cohomology of $G$ with coefficients in the base ring, i.e., $H^*(K(G, n)) \cong H^*(G; R)$

Cohomology of classifying spaces

• Classifying spaces $BG$ are spaces that classify principal $G$-bundles, where $G$ is a topological group
• The Serre spectral sequence can be applied to the universal bundle $G \to EG \to BG$, where $EG$ is a contractible space on which $G$ acts freely
• The spectral sequence shows that the cohomology of the classifying space $BG$ is isomorphic to the group cohomology of $G$ with coefficients in the base ring, i.e., $H^*(BG) \cong H^*(G; R)$
  • This connection between the cohomology of classifying spaces and group cohomology has important applications in algebraic topology and representation theory

Serre spectral sequence in homology

Homological Serre spectral sequence

• The Serre spectral sequence can also be formulated in homology, relating the homology of the base space, fiber, and total space of a fibration
• The $E^2$ page of the homological Serre spectral sequence has terms $E^2_{p,q} = H_p(B; H_q(F))$, involving the homology of the base space with coefficients in the homology of the fiber
• The differentials and convergence properties are analogous to the cohomological version, providing a means to compute the homology of the total space

Comparison with cohomological version

• The homological and cohomological Serre spectral sequences are related by a process called universal coefficient duality
• Under certain conditions (such as the base ring being a field), the homological and cohomological spectral sequences are dual to each other
• This duality allows for the transfer of information between the two versions and provides additional tools for computation and understanding the structure of the spaces involved

Generalization of Serre spectral sequence

Grothendieck spectral sequence

• The Grothendieck spectral sequence is a generalization of the Serre spectral sequence that applies to more general situations, such as compositions of functors
• It relates the derived functors of the composition of two functors to the derived functors of the individual functors
• The Serre spectral sequence can be seen as a special case of the Grothendieck spectral sequence when applied to the composition of the global section functor and the pushforward functor in sheaf cohomology

Spectral sequences in sheaf cohomology

• Spectral sequences play a crucial role in sheaf cohomology, which studies the cohomology of sheaves on topological spaces
• The Leray spectral sequence is a generalization of the Serre spectral sequence in the context of sheaf cohomology, relating the sheaf cohomology of a space to the sheaf cohomology of its image under a continuous map
• Other spectral sequences, such as the Čech-to-derived functor spectral sequence and the hypercohomology spectral sequence, provide powerful tools for computing sheaf cohomology and understanding the structure of sheaves

Examples and calculations

Cohomology of complex projective spaces

• The Serre spectral sequence can be used to compute the cohomology of complex projective spaces $\mathbb{CP}^n$
• By considering the fiber bundle $S^1 \to S^{2n+1} \to \mathbb{CP}^n$, the spectral sequence reveals that the cohomology ring of $\mathbb{CP}^n$ is isomorphic to the truncated polynomial ring $\mathbb{Z}[x]/(x^{n+1})$, where $x$ is a generator of degree 2
• This calculation demonstrates the power of the Serre spectral sequence in determining the cohomology of spaces that arise as fiber bundles

Cohomology of Stiefel manifolds

• Stiefel manifolds $V_k(\mathbb{R}^n)$ are spaces of orthonormal $k$-frames in $\mathbb{R}^n$
• They can be realized as fiber bundles over Grassmann manifolds $G_k(\mathbb{R}^n)$ with fiber $O(k)$, the orthogonal group
• The Serre spectral sequence can be employed to compute the cohomology of Stiefel manifolds by analyzing the cohomology of the base Grassmann manifold and the fiber $O(k)$
  • The differentials in the spectral sequence encode the twisting of the bundle and provide relations between the cohomology classes of the base and fiber

Other notable examples

• The Serre spectral sequence has been successfully applied to compute the cohomology of various other spaces, such as:
  • Flag manifolds, which generalize complex projective spaces and Grassmann manifolds
  • Homogeneous spaces, which are spaces that admit a transitive action by a Lie group
  • Loop spaces, which are spaces of based loops in a topological space
• These examples showcase the versatility and effectiveness of the Serre spectral sequence in understanding the cohomological structure of a wide range of topological spaces

Relationship to other spectral sequences

Comparison with Atiyah-Hirzebruch spectral sequence

• The Atiyah-Hirzebruch spectral sequence is another important spectral sequence in algebraic topology, which relates the singular cohomology of a space to its generalized cohomology theories, such as K-theory or cobordism theory
• While the Serre spectral sequence deals with the cohomology of fibrations, the Atiyah-Hirzebruch spectral sequence is concerned with the relationship between different cohomology theories
• In some cases, the Serre spectral sequence can be used in conjunction with the Atiyah-Hirzebruch spectral sequence to compute generalized cohomology groups of spaces that arise as fiber bundles

Connection to Adams spectral sequence

• The Adams spectral sequence is a powerful tool in stable homotopy theory, which computes the stable homotopy groups of a space or spectrum
• It is constructed using the Ext groups of the Steenrod algebra, which capture important information about the cohomology of the space
• The Serre spectral sequence can be used to compute the cohomology of spaces that appear in the construction of the Adams spectral sequence, such as Eilenberg-MacLane spaces and classifying spaces
  • This connection highlights the interplay between different spectral sequences and their role in understanding the homotopy and cohomology structure of spaces

Limitations and challenges

Computational complexity

• While the Serre spectral sequence provides a systematic approach to computing cohomology, the calculations involved can be computationally complex, especially for spaces with intricate fibration structures
• The number of terms and differentials in the spectral sequence grows rapidly with the dimensions of the base and fiber, making manual computations challenging in higher degrees
• Computational tools and algebraic software have been developed to assist with spectral sequence calculations, but the complexity remains a significant hurdle in many applications

Determining differentials

• One of the main challenges in using the Serre spectral sequence is determining the differentials between the pages of the spectral sequence
• The differentials encode crucial information about the fibration and the cohomology of the total space, but their computation often requires additional tools and insights
• In some cases, the differentials can be determined using geometric arguments, naturality properties, or comparison with known spectral sequences
  • However, in many situations, determining the differentials remains a difficult and open problem

Extension problems

• Even when the Serre spectral sequence converges and provides a filtration of the cohomology of the total space, there may be ambiguities in determining the precise structure of the cohomology groups
• Extension problems arise when the associated graded of the filtration does not completely determine the cohomology ring structure
• Resolving these extension problems often requires additional arguments, such as analyzing the multiplicative structure of the spectral sequence or using secondary operations
  • The presence of extension problems can limit the conclusiveness of the spectral sequence computations and require further investigation