4.1 Introduction to spectral sequences
4 min read•july 30, 2024
Spectral sequences are powerful tools in algebraic topology, helping us compute complex cohomology groups. They consist of pages with differentials, each page refining our understanding of the target cohomology. The is a key example.
This sequence starts with ordinary cohomology and converges to generalized cohomology theories. It's crucial for understanding how different cohomology theories relate, allowing us to compute one theory from another. This makes it a fundamental tool in K-theory calculations.
Spectral sequences: definition and components
Definition and structure
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- Spectral sequences are collections of pages (Er), each consisting of a bigraded module, with differentials dr of bidegree (r,1-r) such that dr ∘ dr = 0
- The pages (Er) of a spectral sequence are indexed by a non-negative integer r, starting with the E0 page
- Each page Er is a differential bigraded module, consisting of a collection of bigraded abelian groups Er(p,q) with a differential dr: Er(p,q) → Er(p+r,q-r+1) satisfying dr ∘ dr = 0
Differentials and cohomology
- The differentials dr on each page have bidegree (r,1-r), mapping elements of Er(p,q) to elements of Er(p+r,q-r+1)
- The cohomology of the differential dr on the Er page gives rise to the next page Er+1, with Er+1(p,q) = H(Er(p,q), dr)
- The differentials and resulting cohomology groups on each page determine the structure and behavior of the spectral sequence
Structure and convergence of spectral sequences
Convergence properties
- A spectral sequence is said to converge to a H if there exists an r0 such that for all r ≥ r0, the differentials dr are zero, and the resulting pages Er are isomorphic to Er0
- The limit term E∞ of a convergent spectral sequence is the graded module obtained by taking the direct sum of the Er0 pages for r ≥ r0
- can be used to compute the graded module H, which often represents a cohomology or homology group of interest
- Spectral sequences can converge conditionally or strongly, depending on the properties of the and the structure of the pages
Relationship with filtrations
- Spectral sequences are often constructed from filtrations, which are increasing sequences of subspaces or submodules
- A filtration of a module M is a sequence of submodules ... ⊂ FpM ⊂ Fp+1M ⊂ ... such that M is the union of all FpM
- The associated graded module of a filtration is defined as GrpM = FpM / Fp-1M, measuring the "jumps" in the filtration
- The E0 page of a spectral sequence associated to a filtration is given by the associated graded module, with E0(p,q) = GrpFp+qM
- The differentials dr of the spectral sequence are induced by the boundary maps in the long associated to the filtration
- The convergence of the spectral sequence is related to properties of the filtration, such as completeness and exhaustiveness
Applications of spectral sequences
Computing cohomology and homology
- Spectral sequences are powerful tools for and homology groups in various algebraic and topological contexts
- The computes the cohomology of a fibration, relating the cohomology of the base space, fiber, and total space
- The computes the cohomology of a pullback or a pushout, based on the cohomology of the input spaces and maps
- The computes stable homotopy groups of spheres and other spaces, based on the cohomology of the space and its Steenrod algebra structure
Establishing long exact sequences
- Spectral sequences can be used to establish the existence of long exact sequences in cohomology and homology
- By studying the structure of the pages and differentials, one can derive long exact sequences relating the cohomology or homology groups of interest
- Examples include the Gysin long exact sequence in cohomology, derived from the Leray-Serre spectral sequence of a sphere bundle, and the Mayer-Vietoris long exact sequence, derived from the Mayer-Vietoris spectral sequence of a cover
Spectral sequences vs filtrations
- Spectral sequences and filtrations are closely related concepts, with spectral sequences often arising from filtrations
- A filtration provides a way to decompose a module into a sequence of submodules, while a spectral sequence studies the relationships between the associated graded pieces
- The construction of a spectral sequence from a filtration involves defining the differentials based on the boundary maps in the long exact sequence of the filtration
- The convergence of the spectral sequence depends on the properties of the filtration, such as completeness (every element is contained in some stage of the filtration) and exhaustiveness (the filtration eventually captures all elements)
- Filtrations can also be used to define other algebraic objects, such as filtered complexes and filtered algebras, which give rise to spectral sequences with additional structure and properties
Key Terms to Review (16)
Adams Spectral Sequence: The Adams spectral sequence is a computational tool used in stable homotopy theory to derive information about the stable homotopy groups of spheres and to compute cohomology theories. This powerful sequence is constructed using the notion of filtrations and is particularly important for understanding various cohomological aspects in algebraic topology.
Atiyah-Hirzebruch Spectral Sequence: The Atiyah-Hirzebruch spectral sequence is a powerful tool in algebraic topology that provides a way to compute the K-theory of a space from its cohomology. It connects the geometry of vector bundles to topological invariants, allowing for the classification of vector bundles through the lens of K-theory and characteristic classes.
Cartan-Eilenberg Theorem: The Cartan-Eilenberg Theorem provides a foundational result in the study of derived functors and spectral sequences, particularly in the context of homological algebra. It establishes an important relationship between the homology of a complex and its derived functors, which can be effectively computed using spectral sequences, linking algebraic structures with topological properties.
Computing cohomology: Computing cohomology refers to the process of determining the cohomology groups of a topological space or algebraic structure, which are vital for understanding its properties and relationships. This involves using various mathematical tools and techniques, such as spectral sequences, to extract information from a complex structure by relating it to simpler ones. The significance of computing cohomology lies in its applications across different areas of mathematics, such as algebraic topology and algebraic geometry.
Computing Homology: Computing homology refers to the process of determining the homology groups of a topological space or a simplicial complex, which are algebraic structures that provide information about the shape and connectivity of the space. These homology groups, denoted as $H_n$, help classify topological spaces by revealing their features like holes and voids in various dimensions, essential for applications in algebraic topology and related fields.
Convergence: Convergence refers to the process by which a sequence of objects or structures approaches a limit or a final state in a mathematical sense. In the context of spectral sequences, convergence indicates that the successive approximations provided by the spectral sequence eventually stabilize, leading to a well-defined object that accurately reflects the homological properties being studied.
D_r: In the context of spectral sequences, $d_r$ refers to the differential at the $r$-th page of the spectral sequence, a crucial component that encodes how information is transmitted between successive pages. These differentials help to construct the cohomological data from the associated filtered complex and reveal important algebraic structures within the spectral sequence. Understanding $d_r$ is essential for interpreting the convergence and behavior of spectral sequences in various mathematical contexts.
Differential: In mathematics, a differential is a fundamental concept that represents an infinitesimal change in a function concerning its variables. This notion is vital for understanding calculus and is essential in various branches of mathematics, particularly in the study of manifolds and algebraic topology.
E_r: The term e_r refers to the r-th component of the universal coefficient spectral sequence, which is a tool in algebraic topology and homological algebra used to compute the homology or cohomology groups of topological spaces or complexes. This component plays a crucial role in relating the spectral sequence to derived functors and is essential for understanding the convergence properties and structure of spectral sequences.
Eilenberg-Moore Spectral Sequence: The Eilenberg-Moore Spectral Sequence is a powerful tool in homological algebra and algebraic topology that arises from a fibration involving a simplicial set or a topological space with a fibration structure. This spectral sequence connects the homology of a space to the homology of its associated fibration, often simplifying complex computations by allowing one to work with layers of abelian groups that are easier to handle. It provides a systematic way to compute the derived functors of functors in the context of sheaf theory and cohomology.
Exact Sequence: An exact sequence is a sequence of algebraic structures and morphisms between them, where the image of one morphism equals the kernel of the next. This concept plays a critical role in connecting various areas of mathematics, particularly in homological algebra and K-Theory, where it helps describe relationships between different objects and their properties.
Filtration: Filtration refers to a systematic method of breaking down a complex object into simpler, more manageable components, often organized hierarchically. In the context of certain mathematical theories, this concept plays a crucial role in analyzing structures by allowing one to study their properties at different levels of granularity, which is especially useful when computing invariants such as K-groups or utilizing advanced tools like spectral sequences.
Graded module: A graded module is a mathematical structure consisting of a module equipped with a grading, which is a decomposition into submodules indexed by integers. This concept is essential for organizing elements according to their 'degree' or 'dimension', allowing for the exploration of algebraic properties and relationships. Graded modules play a critical role in various areas, such as homological algebra and the study of spectral sequences, facilitating the analysis of complex structures by simplifying them into manageable pieces.
Jean-Pierre Serre: Jean-Pierre Serre is a renowned French mathematician known for his significant contributions to topology, algebraic geometry, and number theory. His work laid the foundation for many fundamental results in algebraic K-theory, establishing important connections between various mathematical fields and influencing future research directions.
Leray-Serre Spectral Sequence: The Leray-Serre spectral sequence is a powerful tool in algebraic topology that helps compute the homology groups of a fibration. It provides a way to systematically break down complex spaces into simpler pieces, making it easier to study their topological properties. This spectral sequence arises from the filtration of a space by its fibers and base space, capturing the relationships between their homology groups.
Spectral Sequence Theorem: The Spectral Sequence Theorem is a powerful tool in algebraic topology that provides a method to compute homology or cohomology groups by filtering a complex and organizing the information in a sequence of approximations. It connects different layers of algebraic structures through a series of pages, allowing mathematicians to systematically derive results about topological spaces and their properties, making it essential for studying the relationships between various algebraic invariants.