8.1 Spectral sequences

Spectral sequences are powerful tools for computing homology and cohomology groups in various mathematical fields. They provide a systematic approach to approximating these groups through successive stages, using differential bigraded modules with specific properties.

Key aspects of spectral sequences include convergence, degeneracy, and multiplicativity. These properties determine how effectively the sequence approximates the desired groups, simplifies computations, and provides additional structural information. Understanding these concepts is crucial for applying spectral sequences in diverse mathematical contexts.

Definition of spectral sequences

• Spectral sequences are a powerful algebraic tool used to compute homology and cohomology groups in various mathematical contexts, such as algebraic topology, algebraic geometry, and homological algebra
• They provide a systematic way to approximate the desired homology or cohomology groups through a sequence of successive approximations, known as pages or stages
• Spectral sequences consist of a family of differential bigraded modules $\{E_r^{p,q}\}$, where each $E_r^{p,q}$ is a module equipped with a differential $d_r$ of bidegree $(r, 1-r)$ satisfying $d_r \circ d_r = 0$

Key properties of spectral sequences

Convergence of spectral sequences

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• Convergence is a crucial property that determines whether a spectral sequence successfully approximates the desired homology or cohomology groups
• A spectral sequence is said to converge to a graded module $H^*$ if there exists an $r_0$ such that for all $r \geq r_0$, the modules $E_r^{p,q}$ stabilize and are isomorphic to the associated graded module of a filtration on $H^*$
• Convergence can be influenced by the structure of the spectral sequence and the nature of the filtration used in its construction

Degeneracy of spectral sequences

• Degeneracy occurs when a spectral sequence collapses at a certain page, meaning that all subsequent differentials are zero
• A spectral sequence is said to degenerate at the $E_r$ page if $d_r = 0$ for all $r \geq r_0$, where $r_0$ is some fixed integer
• Degeneracy can simplify the computation of the limit term of the spectral sequence, as it implies that the $E_r$ page is isomorphic to the $E_\infty$ page

Multiplicativity of spectral sequences

• Some spectral sequences possess a multiplicative structure, which means that the pages $E_r$ have a compatible product structure
• Multiplicativity can arise from cup products in cohomology or from the structure of a differential graded algebra
• The presence of a multiplicative structure can provide additional information and help in the computation of the limit term of the spectral sequence

Construction of spectral sequences

Filtrations and spectral sequences

• Spectral sequences are often constructed from a filtered complex, which is a chain complex equipped with a decreasing sequence of subcomplexes
• The filtration induces a spectral sequence by considering the successive quotients of the filtration and their homology groups
• The pages of the spectral sequence are determined by the homology of the associated graded complex of the filtration

Exact couples and spectral sequences

• Exact couples provide another way to construct spectral sequences
• An exact couple consists of a pair of modules $(D, E)$ with maps between them satisfying certain exactness conditions
• The long exact sequence associated with an exact couple gives rise to a spectral sequence, where the pages are determined by the homology of the mapping cone of the differential

Grothendieck's approach to spectral sequences

• Grothendieck introduced a general framework for constructing spectral sequences using the language of derived categories and derived functors
• In this approach, spectral sequences arise from the existence of a filtration on a derived functor
• Grothendieck's formalism provides a unified way to study spectral sequences and their properties in various algebraic and geometric contexts

Important examples of spectral sequences

Serre spectral sequence

• The Serre spectral sequence is a powerful tool in algebraic topology that relates the homology of a fiber bundle to the homology of its base space and fiber
• It is constructed from the filtration of the total space by the preimages of the skeleta of the base space under the projection map
• The Serre spectral sequence has important applications in the computation of homology and cohomology groups of spaces that admit a fibration structure (e.g., sphere bundles, loop spaces)

Atiyah-Hirzebruch spectral sequence

• The Atiyah-Hirzebruch spectral sequence is a generalization of the Serre spectral sequence to the setting of generalized cohomology theories, such as K-theory and cobordism
• It relates the generalized cohomology of a space to its ordinary cohomology and the coefficients of the generalized cohomology theory
• The Atiyah-Hirzebruch spectral sequence has applications in the computation of K-theory and cobordism groups of spaces and in the study of characteristic classes

Adams spectral sequence

• The Adams spectral sequence is a powerful tool in stable homotopy theory that computes the stable homotopy groups of a space or spectrum
• It is constructed using the Adams resolution, which is a projective resolution of the sphere spectrum in the category of modules over the Steenrod algebra
• The Adams spectral sequence has important applications in the computation of stable homotopy groups and in the study of stable homotopy equivalences between spaces or spectra

Eilenberg-Moore spectral sequence

• The Eilenberg-Moore spectral sequence is a cohomological spectral sequence that relates the cohomology of a pullback or fiber product to the cohomology of its factors
• It is constructed using the bar resolution and the associated filtration of the cochain complex of the pullback
• The Eilenberg-Moore spectral sequence has applications in the study of cohomology rings of spaces that arise as pullbacks or fiber products, such as loop spaces and classifying spaces of groups

Applications of spectral sequences

Computation of cohomology groups

• Spectral sequences provide a systematic way to compute cohomology groups of spaces, often by reducing the problem to simpler cohomological computations
• They can be used to compute the cohomology of fiber bundles, loop spaces, classifying spaces, and other spaces that admit a suitable filtration or decomposition
• Spectral sequences can also be used to compute the cohomology of algebraic varieties and schemes in algebraic geometry

Obstruction theory and spectral sequences

• Spectral sequences play a crucial role in obstruction theory, which studies the existence and classification of lifts or extensions of continuous maps between spaces
• Obstruction classes, which measure the obstruction to extending a map, can often be identified with certain differentials in a spectral sequence
• The vanishing of obstruction classes corresponds to the degeneracy of the spectral sequence at a certain page, allowing for the extension of the map

Spectral sequences in algebraic topology

• Spectral sequences are indispensable tools in algebraic topology, where they are used to compute homology, cohomology, and homotopy groups of spaces
• They provide a way to break down complex topological problems into simpler pieces and to study the relationships between different invariants of spaces
• Examples of spectral sequences in algebraic topology include the Serre spectral sequence, the Atiyah-Hirzebruch spectral sequence, and the Adams spectral sequence

Spectral sequences in algebraic geometry

• Spectral sequences also play a significant role in algebraic geometry, where they are used to study the cohomology of algebraic varieties and schemes
• They provide a way to relate the cohomology of a variety to the cohomology of its subvarieties or to the cohomology of its associated sheaves
• Examples of spectral sequences in algebraic geometry include the Hodge spectral sequence, the Leray spectral sequence, and the Grothendieck spectral sequence

Comparison of spectral sequences

Comparison theorems for spectral sequences

• Comparison theorems allow for the comparison of different spectral sequences that compute the same homological or cohomological invariants
• They establish isomorphisms between the pages of the spectral sequences under suitable hypotheses, such as the existence of a map of filtered complexes or a map of exact couples
• Comparison theorems can be used to relate different constructions of spectral sequences and to transfer information between them

Morphisms of spectral sequences

• Morphisms of spectral sequences are maps between the pages of two spectral sequences that are compatible with the differentials
• They arise from maps of filtered complexes or maps of exact couples that induce the spectral sequences
• Morphisms of spectral sequences can be used to study the functoriality of spectral sequences and to establish naturality properties

Naturality of spectral sequences

• Naturality is a desirable property of spectral sequences that ensures their compatibility with maps between the objects they compute
• A spectral sequence is said to be natural if it is functorial with respect to maps of the underlying objects (e.g., spaces, complexes, or modules)
• Naturality allows for the study of induced maps on homology or cohomology through the induced maps on the pages of the spectral sequence

Advanced topics in spectral sequences

Convergence issues and the Frölicher spectral sequence

• Convergence issues can arise in certain spectral sequences, particularly in the context of infinite-dimensional spaces or non-complete filtrations
• The Frölicher spectral sequence is an example of a spectral sequence that does not always converge to the expected cohomology groups
• Studying convergence issues and understanding the limitations of spectral sequences is crucial for their proper application and interpretation

Massey products and spectral sequences

• Massey products are higher-order cohomological operations that generalize the cup product and provide additional structure on cohomology groups
• Spectral sequences can be used to compute Massey products by studying the differentials and their relations on the pages of the spectral sequence
• The presence of non-trivial Massey products can provide insights into the homotopy type of a space and its cohomological structure

Equivariant spectral sequences

• Equivariant spectral sequences are spectral sequences that take into account the action of a group on a space or a complex
• They relate the equivariant cohomology of a space to its ordinary cohomology and the cohomology of the group
• Equivariant spectral sequences have applications in the study of group actions on spaces and in the computation of equivariant invariants

Motivic spectral sequences

• Motivic spectral sequences are spectral sequences that arise in the study of motivic cohomology, which is a cohomology theory for algebraic varieties that takes into account both their algebraic and topological properties
• They relate the motivic cohomology of a variety to its algebraic K-theory and to other cohomological invariants
• Motivic spectral sequences have important applications in the study of algebraic cycles, motives, and the structure of algebraic varieties