Glossary

7.3 Adem relations

5 min read•august 20, 2024

are crucial in understanding cohomology operations. They describe how composites of Steenrod operations can be expressed as linear combinations of other operations, forming a fundamental part of the 's structure.

These relations enable the reduction of arbitrary composites to a standard form, providing a basis for the Steenrod algebra. They're essential for computations in algebraic topology and offer insights into the structure of for various spaces.

Steenrod algebra

  • Fundamental algebraic structure in algebraic topology used to study cohomology operations
  • Consists of operations that act on the cohomology of topological spaces
  • Plays a central role in understanding the structure and properties of cohomology theories

Generators of Steenrod algebra

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  • SqiSq^i generate the Steenrod algebra over F2\mathbb{F}_2
    • Sqi:Hn(X;F2)→Hn+i(X;F2)Sq^i: H^n(X; \mathbb{F}_2) \to H^{n+i}(X; \mathbb{F}_2) is a cohomology operation that raises the degree by ii
  • β\beta and Pi\mathcal{P}^i generate the Steenrod algebra over Fp\mathbb{F}_p for odd primes pp
  • provides an alternative set of generators for the Steenrod algebra

Relations in Steenrod algebra

  • Steenrod operations satisfy certain relations that define the algebraic structure of the Steenrod algebra
  • Adem relations are the most fundamental relations among Steenrod operations
    • Describe how composites of Steenrod operations can be expressed in terms of other Steenrod operations
  • Other relations include the , , and the

Adem relations for Steenrod squares

  • Adem relations express the composition of two Steenrod squares in terms of a linear combination of other Steenrod squares

  • For a<2ba < 2b, the Adem relation for Steenrod squares is: SqaSqb=∑i=0⌊a/2⌋(b−i−1a−2i)Sqa+b−iSqiSq^a Sq^b = \sum_{i=0}^{\lfloor a/2 \rfloor} \binom{b-i-1}{a-2i} Sq^{a+b-i} Sq^i

  • Adem relations allow for the reduction of arbitrary composites of Steenrod squares to a unique standard form

Admissible sequences in Adem relations

  • are sequences of non-negative integers (a1,a2,…,ak)(a_1, a_2, \ldots, a_k) satisfying ai≥2ai+1a_i \geq 2a_{i+1} for all ii
  • Sqa1Sqa2⋯SqakSq^{a_1} Sq^{a_2} \cdots Sq^{a_k} form a basis for the Steenrod algebra
    • Every element in the Steenrod algebra can be uniquely expressed as a linear combination of admissible monomials
  • Adem relations can be used to reduce any monomial of Steenrod squares to a linear combination of admissible monomials

Basis for Steenrod algebra using Adem relations

  • Adem relations imply that the Steenrod algebra has a basis consisting of admissible monomials
  • The Steenrod algebra A2\mathcal{A}_2 over F2\mathbb{F}_2 has a basis given by {Sqa1Sqa2⋯Sqak∣(a1,a2,…,ak) is admissible}\{Sq^{a_1} Sq^{a_2} \cdots Sq^{a_k} \mid (a_1, a_2, \ldots, a_k) \text{ is admissible}\}
  • For odd primes pp, the Steenrod algebra Ap\mathcal{A}_p has a basis consisting of admissible monomials in the Bockstein β\beta and the reduced power operations Pi\mathcal{P}^i

Adem relations for reduced powers

  • Adem relations also hold for the reduced power operations Pi\mathcal{P}^i in the Steenrod algebra over Fp\mathbb{F}_p for odd primes pp

  • For a<pba < pb, the Adem relation for reduced power operations is: PaPb=∑i=0⌊a/p⌋(−1)a+i((p−1)(b−i)−1a−pi)Pa+b−iPi\mathcal{P}^a \mathcal{P}^b = \sum_{i=0}^{\lfloor a/p \rfloor} (-1)^{a+i} \binom{(p-1)(b-i)-1}{a-pi} \mathcal{P}^{a+b-i} \mathcal{P}^i

  • Adem relations for reduced powers allow for the reduction of composites of Pi\mathcal{P}^i operations to a standard form

Adem relations in mod p cohomology

  • Adem relations hold in the mod pp Steenrod algebra Ap\mathcal{A}_p for any prime pp
    • For p=2p=2, the Adem relations involve Steenrod squares SqiSq^i
    • For odd primes pp, the Adem relations involve the Bockstein β\beta and reduced power operations Pi\mathcal{P}^i
  • Adem relations provide a way to compute the action of Steenrod operations on the mod pp cohomology of spaces

Applications of Adem relations

  • Adem relations are a powerful tool in algebraic topology with numerous applications
  • Enable computations in the cohomology of various spaces and provide insights into the structure of cohomology rings
  • Play a crucial role in understanding the behavior of cohomology operations and their properties

Computations using Adem relations

  • Adem relations can be used to compute the action of Steenrod operations on the cohomology of specific spaces
    • Example: Computing the mod 2 cohomology ring of real projective spaces RPn\mathbb{R}P^n using Steenrod squares and Adem relations
  • Adem relations simplify calculations by reducing composites of Steenrod operations to a standard form
  • based on Adem relations have been implemented in computer algebra systems for cohomology calculations

Cohomology of Eilenberg-MacLane spaces

  • K(G,n)K(G, n) are spaces with a single non-trivial homotopy group GG in dimension nn
  • Adem relations can be used to determine the mod pp cohomology rings of Eilenberg-MacLane spaces
    • Example: The mod 2 cohomology of K(Z/2Z,n)K(\mathbb{Z}/2\mathbb{Z}, n) is a polynomial algebra on generators related to Steenrod squares
  • Understanding the cohomology of Eilenberg-MacLane spaces is important in homotopy theory and obstruction theory

Cohomology of classifying spaces

  • BGBG are spaces that classify principal GG-bundles for a topological group GG
  • Adem relations play a role in computing the mod pp cohomology rings of classifying spaces
    • Example: The mod 2 cohomology of the classifying space BO(n)BO(n) for the orthogonal group O(n)O(n) involves Stiefel-Whitney classes and Steenrod squares
  • Cohomology of classifying spaces is connected to and obstruction theory

Adem relations in spectral sequences

  • are algebraic tools used to compute homology and cohomology groups
  • Adem relations can be used to study the behavior of Steenrod operations in spectral sequences
    • Example: The Serre spectral sequence for a fibration can be equipped with Steenrod operations satisfying Adem relations
  • Adem relations provide additional structure and constraints on the differentials and extensions in spectral sequences

Generalizations of Adem relations

  • Adem relations have been generalized and extended to various contexts beyond the classical Steenrod algebra
  • These generalizations capture similar relations among cohomology operations in different settings
  • Generalized Adem relations provide insights into the structure and properties of cohomology theories

Adem relations for Dyer-Lashof operations

  • are homology operations that are dual to Steenrod operations
  • Adem relations for Dyer-Lashof operations describe the composition of these operations
    • Example: For the Dyer-Lashof algebra over F2\mathbb{F}_2, there are Adem relations involving the Dyer-Lashof operations QiQ^i
  • Dyer-Lashof operations and their Adem relations are important in the study of infinite loop spaces and stable homotopy theory

Adem relations in cobordism theory

  • Cobordism theories, such as unoriented cobordism and complex cobordism, have associated cohomology theories
  • Adem relations can be formulated for the cohomology operations in cobordism theories
    • Example: In complex , there are Adem relations involving the Landweber-Novikov operations
  • Cobordism theories and their Adem relations provide powerful tools for studying manifolds and their invariants

Adem relations in other cohomology theories

  • Adem relations have been studied in various beyond ordinary cohomology
    • Example: In K-theory, there are Adem relations for the Steenrod operations acting on K-theory groups
  • Generalized cohomology theories, such as elliptic cohomology and chromatic homotopy theory, also exhibit Adem-type relations among their cohomology operations
  • Investigating Adem relations in these contexts provides insights into the structure and properties of generalized cohomology theories

Key Terms to Review (23)

Adem relations: Adem relations are a set of relations among cohomology operations that describe how to decompose higher operations into simpler ones. They reveal connections between different cohomology classes and show how certain operations can be expressed in terms of others, highlighting the algebraic structure of cohomology theories. Understanding these relations is essential for working with cohomology operations and Wu classes, as they provide insight into the interactions between various cohomological constructs.
Admissible monomials: Admissible monomials are specific algebraic expressions in the context of cohomology theory that satisfy certain combinatorial conditions, allowing them to be used in the calculation of operations like the Adem relations. They are essential in understanding how operations on cohomology classes can be expressed and manipulated, especially in relation to the interaction between different generators.
Admissible sequences: Admissible sequences are specific sequences of indices used in the context of cohomology and homotopy theory, particularly when discussing operations in the Steenrod algebra. These sequences help in defining certain relations and operations that respect the structure of the algebra, which is vital for understanding how different elements interact under specific cohomological operations.
Bockstein Homomorphism: The Bockstein homomorphism is a tool in algebraic topology that arises in the context of cohomology with coefficients in a field or ring. It connects the cohomology groups of a space when moving from a certain coefficient group to a larger one, particularly in situations involving mod p coefficients. This homomorphism plays an essential role in understanding the structure of cohomology rings and can be used to derive Adem relations, which are critical in studying the operations on cohomology classes.
Cartan formula: The Cartan formula is a fundamental equation in cohomology theory that describes the relationship between the cup product and the action of cohomology operations, particularly in relation to Steenrod squares. It provides a way to compute the cohomology of a space by relating it to the structure of the cohomology ring and reveals important interactions between various cohomology operations.
Characteristic classes: Characteristic classes are a way to associate cohomology classes to vector bundles, providing a powerful tool for understanding the geometry and topology of manifolds. They offer insights into the nature of vector bundles, their transformations, and how they relate to the underlying space's topology through cohomological invariants.
Classifying Spaces: Classifying spaces are topological spaces that serve as a universal space for a particular type of bundle, particularly in the context of principal bundles and vector bundles. They encapsulate the properties of the associated bundles, allowing mathematicians to study them via cohomological methods and connect various concepts such as homotopy, cohomology of groups, and characteristic classes.
Cobordism theory: Cobordism theory is a branch of topology that studies the relationships between manifolds through the notion of cobordism, where two manifolds are considered equivalent if there exists a manifold whose boundary is formed by those two manifolds. This theory links various topological invariants and plays a crucial role in classifying manifolds, particularly in terms of their dimensions and structures. It connects with important concepts such as characteristic classes and helps in understanding how different mathematical objects can be related or transformed into one another.
Cohomological Dimension: Cohomological dimension is a concept in algebraic topology that measures the complexity of a space or a module by determining the highest degree of cohomology groups that can be non-zero. This dimension provides insight into how 'complicated' a space is in terms of its topological features and relationships, linking directly to various important theories and results in cohomology.
Cohomology Rings: Cohomology rings are algebraic structures that arise from cohomology groups, where the elements of these groups can be combined using a bilinear operation known as the cup product. This structure captures both topological information about spaces and algebraic relationships between cohomology classes. Cohomology rings are essential for understanding how the properties of a space can be represented in terms of its cohomology groups and products, leading to deeper insights into the algebraic topology of spaces.
Computational techniques: Computational techniques refer to the systematic methods and algorithms used to solve mathematical problems, often with the aid of computers. In the context of algebraic topology, these techniques are particularly valuable for calculating cohomology groups and understanding their structure through various relations. They can facilitate the exploration of complex topological spaces and their associated algebraic invariants.
Dyer-Lashof operations: Dyer-Lashof operations are a set of cohomological operations defined on the cohomology of spaces with a given action of the stable homotopy category, particularly in the context of the mod 2 cohomology ring of a space. These operations help to understand the structure and relationships of stable cohomological phenomena, linking them to other algebraic tools such as the Adem relations, which express how these operations interact with one another.
Eilenberg-MacLane Spaces: Eilenberg-MacLane spaces are topological spaces that classify cohomology theories, characterized by having a single nontrivial homotopy group. Specifically, for each integer n, the space K(G, n) has its nth homotopy group isomorphic to an abelian group G and all other homotopy groups trivial. These spaces play a crucial role in the study of cohomology of spaces, provide examples for cohomology operations, and are essential in understanding Adem relations in algebraic topology.
Generalized cohomology theories: Generalized cohomology theories are a class of cohomology theories that extend the traditional notion of cohomology by providing more flexible and powerful tools for studying topological spaces. They allow mathematicians to analyze various properties of spaces in a broader context, incorporating additional structures like operations and axioms that go beyond ordinary cohomology. This flexibility is crucial when dealing with complex spaces, especially when applying Adem relations, which express relationships among certain cohomology operations.
Homotopy groups: Homotopy groups are algebraic invariants associated with a topological space that measure the space's higher-dimensional holes. They generalize the concept of fundamental groups to higher dimensions, providing important information about the shape and structure of spaces in algebraic topology, particularly in relation to concepts like deformation retraction, stable homotopy, and cohomological techniques.
Hopf Invariant One Theorem: The Hopf Invariant One Theorem is a significant result in algebraic topology that relates the concept of homotopy and cohomology to the topology of spheres. It provides a way to classify certain maps from spheres into other spaces, particularly focusing on the first homotopy group of the target space. This theorem is closely tied to the structure of cohomology rings and plays a crucial role in understanding the Adem relations, which govern the behavior of these cohomological operations.
Instability conditions: Instability conditions are criteria or requirements that determine the lack of stability within certain algebraic structures or cohomological systems. These conditions are important in the context of cohomology, as they can influence the relationships and operations among elements, particularly in relation to Adem relations, which govern how elements interact in a cohomological setting.
Milnor Basis: The Milnor basis is a specific type of basis for the stable homotopy groups of spheres, representing classes of stable cohomology operations. It plays a crucial role in the understanding of how these operations can be expressed in terms of simpler components and their relationships, which connects deeply with the Adem relations in cohomology theory, showcasing the structure and properties of these algebraic operations.
Reduced power operations: Reduced power operations are algebraic operations in the context of cohomology that generalize the idea of taking a power of an element in a cohomological ring, but with a focus on how these operations behave under certain restrictions. They play a significant role in understanding the relationships between different cohomology theories and the structure of the associated rings, particularly in how they interact with classes in these theories.
Spectral Sequences: Spectral sequences are a powerful computational tool in algebraic topology and homological algebra that allow mathematicians to systematically extract information from complex structures. They provide a way to compute homology or cohomology groups by organizing the problem into a series of simpler steps, often transforming a difficult computation into a more manageable form. Spectral sequences are crucial in various areas, including the study of cohomology rings, cohomology operations, and the relationships between different cohomological theories.
Stable cohomology: Stable cohomology refers to the behavior of cohomology groups of spaces as they are 'stabilized' by taking products with a fixed space, typically a sphere. This concept is crucial in understanding how cohomology changes with respect to dimension and plays an important role in connecting various algebraic structures and classes, such as Wu classes and Adem relations, within stable homotopy theory.
Steenrod Algebra: Steenrod Algebra is a mathematical structure that captures the operations of cohomology, particularly in the context of stable homotopy theory. It consists of a collection of cohomological operations known as Steenrod squares, which act on cohomology classes and are used to study the relationships between different cohomology theories. These operations are algebraically rich and are subject to certain relations, which help classify and understand the topology of spaces.
Steenrod squares: Steenrod squares are cohomology operations that act on the cohomology groups of topological spaces, providing a way to understand how these groups behave under certain transformations. They extend the concept of cup products in cohomology, allowing mathematicians to study the relationships between different cohomology classes and gain insights into the topology of the underlying spaces. Steenrod squares also connect to other advanced concepts, such as Wu classes and Stiefel-Whitney classes, creating a rich framework for exploring algebraic topology.
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