5.1 Introduction to spectral sequences

Spectral sequences are powerful tools in algebraic topology, connecting different mathematical structures. They consist of pages of abelian groups arranged in grids, with differentials linking groups across pages. Understanding their convergence properties is crucial for extracting useful information from computations.

Various types of spectral sequences exist, each tailored to specific mathematical problems. From the Serre spectral sequence for fibrations to the Adams spectral sequence in stable homotopy theory, these tools offer insights into complex topological and algebraic structures.

Spectral sequences and their components

Structure and indexing

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• Spectral sequences consist of a sequence of pages, each containing abelian groups or modules arranged in a grid-like pattern
• Pages Er indexed by integer r ≥ 0 or r ≥ 1, depending on convention used
• Groups on each page denoted as $[E_r^{p,q}](https://www.fiveableKeyTerm:e_r^{p,q})$, where p and q represent position in grid
• Initial page (E0 or E1) determined by specific construction or application of sequence

Differentials and relationships between pages

• Differentials connect different groups within same page through homomorphisms $[d_r](https://www.fiveableKeyTerm:d_r): E_r^{p,q} \to E_r^{p+r,q-r+1}$ satisfying $d^2 = 0$
• Relationship between consecutive pages given by $E_{r+1}^{p,q} = \ker(d_r)/\text{im}(d_r)$, where dr is differential on r-th page
• Concept of convergence refers to stabilization of groups $E_r^{p,q}$ for sufficiently large r, leading to E∞ page

Convergence properties of spectral sequences

Types of convergence

• Strong convergence E∞ page determines associated graded object of filtration on limit object, allowing complete reconstruction of limit
• Weak convergence E∞ page provides partial information about limit, potentially leading to extension problems in recovering full structure
• Rate of convergence varies depending on specific spectral sequence and problem at hand (may stabilize quickly or require many pages)

Implications for computations

• Convergence occurs when differentials dr become trivial (zero maps) for sufficiently large r, resulting in isomorphic pages $E_r \cong E_{r+1} \cong \ldots \cong E_\infty$
• E∞ page of convergent spectral sequence provides information about limiting behavior of sequence, often related to object of interest in computation
• Convergence properties crucial for determining effectiveness and applicability of spectral sequence techniques in solving computational problems
• Understanding convergence allows development of strategies to extract useful information from spectral sequences, even in cases where complete convergence may not occur (partial convergence)

Constructing examples of spectral sequences

Fundamental spectral sequences

• Serre spectral sequence relates homology of total space to homology of base and fiber in fibration (fiber bundles)
• Leray-Serre spectral sequence generalizes Serre spectral sequence to cohomology and incorporates local coefficient systems (twisted cohomology)
• Lyndon-Hochschild-Serre spectral sequence used in group cohomology to relate cohomology of group to normal subgroup and quotient group (group extensions)
• Atiyah-Hirzebruch spectral sequence connects singular cohomology to generalized cohomology theories (K-theory, cobordism)
• Adams spectral sequence powerful tool in stable homotopy theory, used to compute stable homotopy groups of spheres and other spaces

Interpreting spectral sequence structure

• Analyze initial pages to identify key differentials and determine convergence properties
• Recognize patterns and symmetries in arrangement of groups and differentials to gain insights into underlying algebraic or topological structures
• Examine behavior of differentials across pages to understand how information propagates through sequence
• Identify edge homomorphisms and spectral sequence collapse to simplify computations and extract relevant information

Applications of spectral sequences in topology and algebra

Computational applications

• Compute homology and cohomology groups of complex spaces by breaking them down into simpler components (cellular homology)
• Determine homology of fiber bundles and analyze topology of associated total spaces using Serre spectral sequence (Hopf fibration)
• Study cohomology with local coefficients and investigate twisted cohomology theories using Leray-Serre spectral sequence (Möbius strip)
• Compute group cohomology and analyze group extensions using Lyndon-Hochschild-Serre spectral sequence (central extensions)
• Relate different cohomology theories and study their relationships using Atiyah-Hirzebruch spectral sequence (K-theory of spheres)

Theoretical applications

• Prove theorems in algebraic topology using spectral sequence techniques (Hurewicz theorem, Freudenthal suspension theorem)
• Develop and analyze spectral sequences arising from filtered complexes in homological algebra to study derived functors
• Compute Ext and Tor groups using spectral sequences (universal coefficient theorem)
• Investigate relationships between different cohomology theories and their properties (Chern character)
• Study algebraic structures of spectral sequences themselves to gain insights into underlying mathematical objects (motivic cohomology)