Glossary

8.4 Applications in homological algebra

2 min read•august 7, 2024

Spectral sequences are powerful tools in homological algebra, connecting different theories. They allow us to compute complex structures by breaking them down into simpler pieces and using successive approximations.

From Leray and Serre to Eilenberg-Moore and Adams, these sequences tackle various algebraic and topological problems. They're essential for understanding , fibrations, and even of spheres.

Spectral Sequences in Homological Algebra

Key Spectral Sequences for Homology and Cohomology

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  • relates the cohomology of a space with the cohomology of a on that space
    • Enables computing using a F→E→BF \to E \to B by successive approximations
    • Starts with E2p,q=Hp(B;Hq(F))E_2^{p,q} = H^p(B; H^q(F)) and converges to Hp+q(E)H^{p+q}(E)
  • is a that relates the of the base space and the homology of the fiber to the homology of the total space in a fibration
    • For a fibration F→E→BF \to E \to B, it has Ep,q2=Hp(B;Hq(F))E^2_{p,q} = H_p(B; H_q(F)) and converges to Hp+q(E)H_{p+q}(E)
    • Powerful tool for computing of a space when it's realized as a fibration (Hopf fibration of spheres)
  • relates a of a space to its
    • For a spectrum hh and CW complex XX, it starts with E2p,q=Hp(X;πq(h))E_2^{p,q} = H^p(X; \pi_q(h)) and converges to hp+q(X)h^{p+q}(X)
    • Allows computing groups like or from ordinary cohomology (stable homotopy groups of spheres)

Spectral Sequences for Ring and Module Structures

  • relates the cohomology of a fiber space to the cohomology of the base and total space for a fibration of spaces with additional structure
    • Applies when cohomology of fiber, total space and base have compatible ring or module structures
    • Starts with E2p,q=Torp,qH∗(B)(H∗(E),k)E_2^{p,q} = \text{Tor}^{H^*(B)}_{p,q}(H^*(E), k) and converges to Hp+q(F)H^{p+q}(F) as modules over H∗(B)H^*(B)
  • computes stable homotopy groups of spheres from Ext groups in the category of modules over the
    • E2s,t=ExtAs,t(Fp,Fp)⇒πt−sS(S0)(p)E_2^{s,t} = \text{Ext}_{\mathcal{A}}^{s,t}(\mathbb{F}_p, \mathbb{F}_p) \Rightarrow \pi_{t-s}^S(\mathbb{S}^0)_{(p)}
    • Connects stable homotopy theory to homological algebra of the Steenrod algebra ( encoding cohomology operations)

Group Cohomology Spectral Sequences

Spectral Sequences for Group Extensions

  • relates the cohomology of a group extension to the cohomology of the subgroup and quotient group
    • For a group extension 1→H→G→K→11 \to H \to G \to K \to 1 it has E2p,q=Hp(K;Hq(H;M))E_2^{p,q} = H^p(K; H^q(H; M)) converging to Hp+q(G;M)H^{p+q}(G; M)
    • Allows computing group cohomology using the cohomology of smaller pieces HH and KK (when GG is a semidirect product)
  • is a specific case of the Hochschild-Serre spectral sequence
    • Applies to group cohomology with coefficients in a trivial module
    • Has E2p,q=Hp(K;Hq(H;Z))E_2^{p,q} = H^p(K; H^q(H; \mathbb{Z})) and converges to Hp+q(G;Z)H^{p+q}(G; \mathbb{Z})
    • Useful for analyzing the cohomological dimension of solvable groups (successive extensions by abelian groups)

Key Terms to Review (23)

Adams Spectral Sequence: The Adams Spectral Sequence is a powerful computational tool in homological algebra that helps in the study of stable homotopy groups of spheres and related structures. It provides a way to compute these groups by filtering them through a sequence of approximations that converge to the desired results, allowing for detailed calculations in topology. This spectral sequence connects algebraic and topological data, making it invaluable in applications across various areas of mathematics.
Atiyah-Hirzebruch spectral sequence: The Atiyah-Hirzebruch spectral sequence is a powerful tool in algebraic topology and homological algebra that relates the homology of a topological space to the homology of its associated bundles. This spectral sequence provides a systematic way to compute invariants by filtering complexes and organizing information about different layers of the underlying algebraic structure. It plays a crucial role in connecting various cohomology theories and has applications in understanding more complex geometric and topological properties.
Cobordism: Cobordism is a concept in topology that relates to the idea of manifolds being boundaries of higher-dimensional manifolds. Two manifolds are cobordant if there exists a manifold whose boundary is the disjoint union of the two manifolds, which indicates a deep relationship between their geometric properties and dimensions. This concept plays a crucial role in the study of algebraic topology and has important implications in homological algebra, particularly in the classification of manifolds and in linking different algebraic structures.
Cohomology: Cohomology is a mathematical concept that assigns algebraic invariants to topological spaces and chain complexes, capturing information about their structure and relationships. It provides a dual perspective to homology, focusing on the study of cochains and cocycles, which can reveal properties of spaces that homology alone might miss. This tool is essential in various areas of mathematics, connecting geometry, algebra, and topology.
Cohomology Groups: Cohomology groups are algebraic structures that provide a way to study topological spaces and their properties through the lens of algebra. They are defined as the duals of homology groups and capture information about the shape and structure of spaces, making them crucial for understanding various mathematical concepts in both algebra and topology.
Eilenberg-Moore Spectral Sequence: The Eilenberg-Moore Spectral Sequence is a powerful tool in homological algebra that arises from the study of fibrations and the homotopy theory of diagrams. It provides a systematic way to compute the homology of a fibration by associating it with a spectral sequence, which can reveal information about both the base and the fiber. This concept connects deeply to various areas, including algebraic topology and the broader framework of homological methods in mathematics.
Extraordinary Cohomology: Extraordinary cohomology is a type of cohomology theory that extends traditional cohomology theories, such as singular cohomology or de Rham cohomology, to capture additional topological features of spaces. It is particularly significant in the context of algebraic topology, where it can provide deeper insights into the structure of manifolds and schemes, linking them to various algebraic and geometric properties.
Fibration: A fibration is a specific type of mapping in the context of homotopy theory and algebraic topology that captures how one space can be continuously 'projected' onto another. It helps in understanding the structure of spaces through fibers, which are pre-images of points under this mapping. The concept connects deeply with homological algebra by facilitating the study of derived functors and the relationship between different categories.
Generalized cohomology theory: Generalized cohomology theory is a broad framework in algebraic topology that extends the traditional notion of cohomology, allowing for the definition of cohomological invariants in a more flexible way. It provides a way to classify topological spaces using various types of cohomology theories, such as singular cohomology, de Rham cohomology, and others, each capturing different geometric and algebraic properties. This flexibility is essential for applications in homological algebra, where one can study the relationships between different algebraic structures through the lens of these cohomological invariants.
Graded Hopf algebra: A graded Hopf algebra is a type of algebraic structure that combines the properties of both a Hopf algebra and a graded algebra. It consists of a direct sum of vector spaces, each associated with a non-negative integer grade, where the multiplication and comultiplication operations respect this grading. This structure plays a vital role in the study of homological algebra, particularly in areas like cohomology and deformation theory.
Group Cohomology: Group cohomology is a mathematical tool used to study the properties of groups through the lens of cohomological methods, providing insights into their representations and extensions. It generalizes group homology by associating a sequence of abelian groups or modules to a given group and a coefficient module, allowing the analysis of group actions and how they relate to algebraic topology and homological algebra.
Hochschild-Serre Spectral Sequence: The Hochschild-Serre spectral sequence is a powerful tool in homological algebra that arises from a short exact sequence of groups, linking the homology of a group and its normal subgroup to the homology of the quotient group. This spectral sequence provides a way to compute the homology of a group by considering the homology of its subgroups and the interactions between them, making it essential for understanding extensions and cohomological properties.
Homology: Homology is a mathematical concept that associates a sequence of algebraic objects, typically abelian groups or vector spaces, to a topological space or a chain complex, providing a way to classify and measure the 'holes' or 'voids' within that space. It connects deeply with various structures in mathematics, revealing relationships between algebra and topology through its formulation and applications.
Homology groups: Homology groups are algebraic structures that arise from chain complexes and serve to classify topological spaces by measuring their 'holes' in various dimensions. They provide crucial insights into the properties of spaces and are integral to understanding concepts in algebra, geometry, and topology.
K-theory: K-theory is a branch of mathematics that studies vector bundles and their generalizations through a homotopical and categorical lens. It provides powerful tools for understanding the structure of vector bundles over topological spaces, allowing mathematicians to connect algebraic and topological concepts. K-theory plays a significant role in various areas such as algebraic topology, algebraic geometry, and even number theory, making it an essential concept in advanced mathematical studies.
Leray Spectral Sequence: The Leray spectral sequence is a powerful tool in algebraic topology and homological algebra that provides a way to compute homology groups of a topological space by analyzing the structure of a fibration. It connects the homology of a total space, base space, and fiber, effectively allowing one to understand complex spaces through simpler ones. This concept is crucial for working with filtered complexes, double complexes, and has important applications in deriving significant results in homological algebra.
Lyndon-Hochschild-Serre spectral sequence: The Lyndon-Hochschild-Serre spectral sequence is a powerful tool in homological algebra and algebraic topology that arises from a filtered complex, particularly in the context of group extensions. It connects the cohomology of a group with the cohomology of its normal subgroups and the quotient, allowing for computations in group cohomology. This sequence is particularly useful in providing a way to compute the cohomology of groups that can be expressed in terms of subgroups.
Ordinary cohomology: Ordinary cohomology is a mathematical tool that associates a sequence of abelian groups or vector spaces with a topological space, allowing for the study of its properties through algebraic means. It provides insights into the structure and features of spaces by capturing information about their holes and topological features, often revealing relationships between different spaces. This concept plays a significant role in homological algebra, linking algebraic invariants to geometric intuition.
Serre spectral sequence: The Serre spectral sequence is a powerful computational tool in algebraic topology that arises in the study of fibrations, allowing one to compute the homology or cohomology of a space based on the homological properties of its base and fiber. This sequence organizes information about the relationships between different layers of a fibration, making it easier to analyze complex topological spaces. It highlights connections between algebraic structures and topological features, bridging various areas of mathematics.
Sheaf: A sheaf is a mathematical structure that associates data to the open sets of a topological space in a way that locally resembles a function but satisfies certain gluing conditions. It captures the idea of local data and how it can be pieced together to form global information, making it essential in areas such as algebraic geometry and homological algebra.
Spectral sequence: A spectral sequence is a mathematical tool that allows one to compute homology or cohomology groups by systematically breaking down complex objects into simpler pieces. It is built from a sequence of approximations that converge to a desired object, providing a way to handle filtered complexes and understand their properties through successive stages of computation. This method finds significant applications in various areas, including homological algebra and sheaf cohomology.
Stable Homotopy Groups: Stable homotopy groups are a sequence of groups that capture the 'stable' features of topological spaces and their mappings. They emerge from the study of homotopy theory, particularly in relation to spectra, where the notion of stability comes into play when considering maps between spaces as one takes limits or colimits of these structures. This concept is significant because it helps understand the relationships between different topological spaces and provides insight into the algebraic structures that arise from homotopical considerations.
Steenrod Algebra: Steenrod algebra is a mathematical structure that arises in the context of stable homotopy theory, primarily dealing with operations on cohomology classes. It provides a way to understand how cohomological operations interact and is particularly significant in the study of the cohomology of topological spaces. This algebra captures essential information about how different cohomological elements can be combined and manipulated, which is vital for various applications in algebraic topology and homological algebra.
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