🧬Cohomology Theory Unit 8 – Spectral sequences

What's the Big Idea?

• Spectral sequences provide a powerful tool for computing homology and cohomology groups in various contexts
• Break down complex algebraic structures into simpler pieces that can be analyzed step by step
• Consist of a sequence of pages, each containing a differential (a map satisfying $d^2 = 0$) and a resulting homology
• Successive pages reveal more information about the original structure until reaching a stable limit
• Enable the computation of invariants that would otherwise be difficult or impossible to calculate directly
• Arise naturally in many areas of mathematics, including algebraic topology, algebraic geometry, and representation theory
  • Particularly useful for studying the cohomology of fiber bundles and the homology of group actions
• Provide a systematic way to approach problems involving long exact sequences and filtrations

Key Concepts and Definitions

• Differential: A map $d$ satisfying $d^2 = 0$, which generalizes the notion of a boundary map in chain complexes
  • Differentials on each page of the spectral sequence are denoted by $d_r$, where $r$ is the page number
• Filtration: A sequence of subspaces or submodules $F_0 \subseteq F_1 \subseteq \cdots \subseteq F_n$ of a given space or module
  • Filtrations give rise to spectral sequences by considering the associated graded objects
• Convergence: A spectral sequence is said to converge to a limit term, typically denoted by $E_\infty$, which encodes the desired homology or cohomology information
  • Convergence can be conditional or strong, depending on the properties of the filtration and the limit term
• Degeneracy: A spectral sequence is called degenerate if all differentials on a certain page and beyond are zero
  • Degenerate spectral sequences converge quickly but may provide less information than non-degenerate ones
• Spectral sequence of a filtered complex: A spectral sequence arising from a chain complex equipped with a filtration
  • The $E_0$ page consists of the associated graded complex, and subsequent pages are obtained by taking homology with respect to the induced differentials
• Grothendieck spectral sequence: A spectral sequence relating the derived functors of a composite functor to the derived functors of its components
  • Generalizes the Leray spectral sequence and has numerous applications in homological algebra and algebraic geometry

Types of Spectral Sequences

• Serre spectral sequence: Relates the cohomology of a fiber bundle to the cohomology of its base space and fiber
  • Arises from the filtration of the total space by the preimages of an open cover of the base
  • Converges to the cohomology of the total space with coefficients in a given sheaf
• Leray spectral sequence: A generalization of the Serre spectral sequence for sheaf cohomology
  • Relates the sheaf cohomology of a space with coefficients in a given sheaf to the sheaf cohomology of a covering space with coefficients in the direct image sheaf
• Eilenberg-Moore spectral sequence: Computes the homology or cohomology of a pullback or pushout in terms of the homology or cohomology of its components
  • Useful for studying fiber squares and homotopy pullbacks in algebraic topology
• Adams spectral sequence: A tool for computing stable homotopy groups of spheres and other spaces
  • Arises from the Adams resolution, which approximates a space by a tower of fibrations involving Eilenberg-MacLane spaces
• Hypercohomology spectral sequence: Relates the hypercohomology of a complex of sheaves to the cohomology of the individual sheaves and their derived functors
  • Generalizes the Leray spectral sequence and is useful for studying derived categories and sheaf cohomology
• Lyndon-Hochschild-Serre spectral sequence: Computes the group cohomology of an extension of groups in terms of the cohomology of the quotient and the cohomology of the kernel with coefficients in the induced module
  • Arises from the filtration of the group by the powers of the kernel subgroup

How Spectral Sequences Work

• Start with an algebraic structure equipped with a filtration, such as a filtered complex or a sheaf on a filtered space
• The $E_0$ page of the spectral sequence consists of the associated graded objects, which are simpler than the original structure
  • For a filtered complex, the $E_0$ page is the associated graded complex
  • For a sheaf on a filtered space, the $E_0$ page consists of the sheaves on the successive quotients of the filtration
• Each page $E_r$ of the spectral sequence comes with a differential $d_r$, which is a map satisfying $d_r^2 = 0$
  • The differential $d_r$ has bidegree $(r, 1-r)$, meaning it maps $E_r^{p,q}$ to $E_r^{p+r,q-r+1}$
• The next page $E_{r+1}$ is obtained by taking the homology of $E_r$ with respect to the differential $d_r$
  • $E_{r+1}^{p,q} = \ker(d_r: E_r^{p,q} \to E_r^{p+r,q-r+1}) / \operatorname{im}(d_r: E_r^{p-r,q+r-1} \to E_r^{p,q})$
• The process continues, with each page revealing more information about the original structure
  • The differentials on each page are induced by the differentials on the previous page and the filtration
• Under favorable conditions, the spectral sequence converges to a limit term $E_\infty$, which is the associated graded object of the desired homology or cohomology
  • Convergence means that for each $(p,q)$, there exists an $r_0$ such that $E_r^{p,q} \cong E_\infty^{p,q}$ for all $r \geq r_0$
• The limit term $E_\infty$ is often not the actual homology or cohomology but rather a graded version of it
  • To recover the true homology or cohomology, one needs to solve extension problems arising from the filtration

Applications in Cohomology Theory

• Computing the cohomology of fiber bundles: The Serre spectral sequence relates the cohomology of a fiber bundle to the cohomology of its base space and fiber
  • Useful for studying the topology of vector bundles, principal bundles, and other fiber bundles arising in geometry and physics
• Calculating the cohomology of sheaves: The Leray spectral sequence and its generalizations provide a way to compute sheaf cohomology using coverings and direct image sheaves
  • Important for understanding the global properties of sheaves and their derived categories
• Investigating the cohomology of groups: The Lyndon-Hochschild-Serre spectral sequence relates the cohomology of a group extension to the cohomology of the quotient and kernel groups
  • Helps in studying group extensions, group actions, and the structure of cohomology rings
• Analyzing the cohomology of algebraic varieties: Spectral sequences arising from filtrations of algebraic varieties by subvarieties or stratifications can provide insight into their cohomological properties
  • Used in the study of perverse sheaves, intersection cohomology, and the decomposition theorem
• Understanding the relationship between different cohomology theories: Spectral sequences can relate different cohomology theories, such as de Rham cohomology, étale cohomology, and crystalline cohomology
  • Facilitates the comparison of cohomological invariants across different settings and the transfer of results between them

Computational Techniques

• Identify the relevant filtration or algebraic structure giving rise to the spectral sequence
  • This could be a filtered complex, a sheaf on a filtered space, or a group extension, among others
• Determine the $E_0$ page of the spectral sequence by computing the associated graded objects
  • For a filtered complex, this means taking the quotients of successive filtration stages
  • For a sheaf on a filtered space, this involves considering the sheaves on the quotients of the filtration
• Calculate the differentials on each page of the spectral sequence
  • The differentials are induced by the differentials on the original complex or the maps between the successive quotients of the filtration
  • Use the fact that the differentials satisfy $d_r^2 = 0$ and have bidegree $(r, 1-r)$
• Compute the homology of each page with respect to its differential to obtain the next page
  • This involves finding the kernel and image of the differentials and taking their quotient
• Repeat the process until the spectral sequence degenerates or converges to a limit term
  • Convergence occurs when the differentials on a certain page and beyond are all zero
• Interpret the limit term $E_\infty$ in terms of the desired cohomological information
  • The limit term is often a graded version of the actual cohomology, and extension problems may need to be solved to recover the true cohomology
• Use the structure of the spectral sequence and the knowledge of the input data to deduce additional information
  • For example, the vanishing of certain terms or differentials can lead to simplifications or exact sequences relating the cohomology groups of interest

Common Pitfalls and Tips

• Be careful with the indexing and bidegrees of the differentials
  • The differentials $d_r$ have bidegree $(r, 1-r)$, which can be confusing at first
  • Pay attention to the direction of the arrows and the shifting of indices when computing homology
• Keep track of the filtration and its compatibility with the algebraic structures involved
  • The filtration should be compatible with the differentials on the original complex or the maps between sheaves
  • Inconsistencies in the filtration can lead to incorrect results or a lack of convergence
• Be aware of the limitations of spectral sequences
  • Spectral sequences provide a powerful computational tool, but they may not always give complete information
  • The limit term $E_\infty$ may be a graded version of the desired cohomology, and extension problems may need to be solved separately
• Use the structure of the spectral sequence to your advantage
  • The vanishing of certain terms or differentials can simplify computations and lead to exact sequences
  • Symmetries or periodicities in the spectral sequence can also provide valuable insights
• Compare with known results or use universal coefficients theorems when possible
  • Spectral sequence computations can often be checked against known results for specific cases
  • Universal coefficients theorems can help relate the cohomology with different coefficient modules
• Break down complex problems into manageable steps
  • Spectral sequences can be intimidating at first, but they become more tractable when broken down into smaller subproblems
  • Focus on computing one differential or one page at a time, and gradually build up to the overall picture

Real-world Examples and Cool Stuff

• The Serre spectral sequence was used by Jean-Pierre Serre to prove that the higher homotopy groups of spheres are torsion
  • This result has important implications for the classification of topological spaces and the study of stable homotopy theory
• The Leray spectral sequence plays a crucial role in the proof of the Weil conjectures, which relate the topology of algebraic varieties over finite fields to their arithmetic properties
  • The Weil conjectures have deep connections to number theory, algebraic geometry, and representation theory
• The Adams spectral sequence is a central tool in the computation of stable homotopy groups of spheres, which are among the most fundamental and mysterious objects in algebraic topology
  • The stable homotopy groups of spheres exhibit intricate patterns and have been the subject of extensive research for decades
• Spectral sequences have been used to study the cohomology of classifying spaces of groups, which encode important information about the structure and representation theory of the groups
  • The cohomology of classifying spaces is related to characteristic classes, obstruction theory, and the classification of principal bundles
• In physics, spectral sequences have found applications in the study of anomalies, quantum field theories, and string theory
  • For example, the Atiyah-Hirzebruch spectral sequence relates the K-theory of a space to its ordinary cohomology and has been used to study D-brane charges and RR-fields in string theory
• Spectral sequences have also been employed in the study of operads, which are algebraic structures encoding operations with multiple inputs and one output
  • Operads have applications in homotopy theory, algebraic geometry, and mathematical physics, and spectral sequences can help compute their homology and cohomology