Glossary

7.1 Cohomology operations

7 min read•august 20, 2024

are powerful tools in algebraic topology that reveal relationships between cohomology groups. They provide additional structure to cohomology, allowing us to study connections between different dimensions and construct .

These operations, like , have crucial properties such as and . They play key roles in various areas of mathematics, including , , and homotopy theory, enhancing our understanding of topological spaces.

Definition of cohomology operations

  • Cohomology operations are natural transformations between cohomology functors that provide additional structure and information about the cohomology groups of a topological space
  • They allow for the study of relationships between cohomology classes in different dimensions and can be used to construct invariants of topological spaces
  • Cohomology operations are an essential tool in algebraic topology and have applications in various areas of mathematics, including characteristic classes, obstruction theory, and homotopy theory

Properties of cohomology operations

Naturality of cohomology operations

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  • Cohomology operations commute with the induced homomorphisms on cohomology groups arising from continuous maps between topological spaces
  • This naturality property ensures that cohomology operations are well-defined and independent of the choice of representative cocycles for cohomology classes
  • Naturality allows for the comparison of cohomology operations across different spaces and maps, making them a powerful tool in the study of topological invariants

Additivity of cohomology operations

  • Cohomology operations are additive, meaning they preserve the group structure of cohomology groups
  • For any two cohomology classes α\alpha and β\beta, a cohomology operation ϕ\phi satisfies ϕ(α+β)=ϕ(α)+ϕ(β)\phi(\alpha + \beta) = \phi(\alpha) + \phi(\beta)
  • Additivity ensures that cohomology operations are compatible with the algebraic structure of cohomology groups and allows for the study of their behavior under linear combinations of cohomology classes

Cohomology operations on products

Künneth formula

  • The describes the relationship between the cohomology groups of a product space and the cohomology groups of its factors
  • It states that, under certain conditions, the cohomology of a product space is isomorphic to the tensor product of the cohomology groups of its factors
  • Cohomology operations can be defined on product spaces using the Künneth formula, allowing for the study of their behavior under products and the construction of new cohomology classes

Cross product

  • The is a cohomology operation that takes two cohomology classes on separate spaces and produces a cohomology class on their product space
  • Given cohomology classes α∈Hp(X)\alpha \in H^p(X) and β∈Hq(Y)\beta \in H^q(Y), their cross product is a class α×β∈Hp+q(X×Y)\alpha \times \beta \in H^{p+q}(X \times Y)
  • The cross product is natural with respect to maps between spaces and satisfies the Künneth formula, making it a fundamental tool in the study of cohomology operations on product spaces

Steenrod squares

Definition of Steenrod squares

  • Steenrod squares are cohomology operations in mod 2 cohomology that generalize the and provide additional structure on the cohomology groups
  • For each non-negative integer ii, the ii-th Steenrod square is a natural transformation Sqi:Hn(X;Z/2)→Hn+i(X;Z/2)Sq^i: H^n(X; \mathbb{Z}/2) \to H^{n+i}(X; \mathbb{Z}/2)
  • Steenrod squares are essential in the study of characteristic classes, obstruction theory, and the

Properties of Steenrod squares

  • Steenrod squares satisfy several important properties, including naturality, additivity, and the
  • They are , meaning they commute with the suspension isomorphism in cohomology
  • Steenrod squares also satisfy the , which provide a set of relations between compositions of Steenrod squares and allow for their efficient computation

Adem relations

  • The Adem relations are a set of identities that express the composition of two Steenrod squares as a linear combination of other Steenrod squares
  • They provide a way to simplify and compute compositions of Steenrod squares, reducing them to a standard form
  • The Adem relations are essential in the study of the algebraic structure of the Steenrod algebra, which is the algebra of stable cohomology operations in mod 2 cohomology

Cartan formula

  • The Cartan formula describes the behavior of Steenrod squares with respect to the cup product in cohomology
  • It states that for any two cohomology classes α\alpha and β\beta, the Steenrod square of their cup product can be expressed as a sum of cup products of their Steenrod squares
  • The Cartan formula is a powerful tool in the computation of Steenrod squares and the study of their relationship with the multiplicative structure of cohomology

Cohomology operations in spectral sequences

Naturality in spectral sequences

  • Cohomology operations are natural with respect to the differentials and isomorphisms in a spectral sequence
  • This naturality property allows for the study of the behavior of cohomology operations in the context of and the computation of cohomology groups using these operations
  • Naturality in spectral sequences is particularly useful in the study of the Serre spectral sequence and the Eilenberg-Moore spectral sequence, where cohomology operations can provide additional information about the cohomology of fiber bundles and loop spaces

Cohomology operations on the E2E_2 page

  • In many spectral sequences, such as the Serre spectral sequence, the E2E_2 page can be identified with a certain or a tensor product of cohomology groups
  • Cohomology operations can be defined on the E2E_2 page of these spectral sequences, providing additional structure and allowing for the study of the behavior of these operations under the differentials
  • The study of cohomology operations on the E2E_2 page can lead to important results in homotopy theory, such as the computation of the mod 2 cohomology of and the classification of maps between them

Applications of cohomology operations

Characteristic classes

  • Characteristic classes are invariants associated with vector bundles and principal bundles that live in the cohomology of the base space
  • Cohomology operations, particularly Steenrod squares, play a crucial role in the construction and study of characteristic classes, such as Stiefel-Whitney classes, Chern classes, and Pontryagin classes
  • The behavior of characteristic classes under cohomology operations provides important information about the topology of the base space and the structure of the associated bundles

Obstruction theory

  • Obstruction theory is a technique in algebraic topology that uses cohomology groups to study the existence and classification of continuous maps between topological spaces
  • Cohomology operations can be used to construct obstruction classes that live in certain cohomology groups and provide information about the existence and properties of continuous maps
  • The vanishing of certain cohomology operations on obstruction classes can lead to the existence of continuous maps with desired properties, such as lifts or extensions of maps

Homotopy groups of spheres

  • The homotopy groups of spheres are important invariants in algebraic topology that capture information about the higher-dimensional structure of spheres
  • Cohomology operations, particularly Steenrod squares, have been used to study the homotopy groups of spheres and to prove important results, such as the Hopf invariant one theorem and the Freudenthal suspension theorem
  • The relationship between cohomology operations and the homotopy groups of spheres has led to significant advances in the understanding of stable homotopy theory and the computation of stable homotopy groups

Cohomology operations in generalized cohomology theories

Stable cohomology operations

  • Stable cohomology operations are natural transformations between generalized cohomology theories that commute with the suspension isomorphism
  • Examples of stable cohomology operations include the Steenrod squares in mod 2 cohomology and the Steenrod reduced powers in mod p cohomology
  • The study of stable cohomology operations has led to important results in stable homotopy theory, such as the classification of stable cohomology operations and the construction of the Adams spectral sequence

Unstable cohomology operations

  • are natural transformations between generalized cohomology theories that do not necessarily commute with the suspension isomorphism
  • Examples of unstable cohomology operations include the Steenrod squares in and the Dyer-Lashof operations in homology
  • The study of unstable cohomology operations has led to important results in unstable homotopy theory, such as the classification of unstable cohomology operations and the construction of the unstable Adams spectral sequence

Relationship between cohomology operations and homotopy theory

Cohomology operations as homotopy classes of maps

  • Cohomology operations can be interpreted as homotopy classes of maps between certain topological spaces, such as Eilenberg-MacLane spaces
  • This interpretation provides a geometric understanding of cohomology operations and allows for the study of their properties using techniques from homotopy theory
  • The relationship between cohomology operations and homotopy classes of maps has led to important results, such as the classification of stable cohomology operations and the construction of the Steenrod algebra

Cohomology operations and infinite loop spaces

  • Infinite loop spaces are topological spaces with a compatible sequence of deloopings, forming a spectrum in the sense of stable homotopy theory
  • Cohomology operations, particularly stable cohomology operations, can be studied in the context of infinite loop spaces and spectra, providing a rich algebraic and homotopical structure
  • The relationship between cohomology operations and infinite loop spaces has led to important results in stable homotopy theory, such as the Dyer-Lashof algebra and the Nishida relations in the homology of infinite loop spaces

Key Terms to Review (24)

Additivity: Additivity refers to the property in algebraic structures where the operation applied to a combination of elements yields the same result as applying the operation separately and then combining the results. This concept plays a critical role in various mathematical frameworks, particularly in understanding how different spaces or complexes relate to one another through homology and cohomology theories.
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.
Cap Product: The cap product is a fundamental operation in algebraic topology that combines elements from homology and cohomology theories to produce a new cohomology class. This operation helps connect the topological structure of a space with its algebraic properties, allowing for deeper insights into how different dimensions interact within that space.
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.
Cochain Complex: A cochain complex is a sequence of abelian groups or modules connected by homomorphisms, where the composition of consecutive homomorphisms is zero. It serves as a crucial structure in cohomology theory, enabling the computation of cohomology groups that capture topological features of spaces. The relationship between cochain complexes and simplicial complexes highlights how geometric data can translate into algebraic invariants.
Cohomology Group: A cohomology group is a mathematical structure that captures information about the shape and features of a topological space, providing a dual perspective to homology groups. It serves as an algebraic tool to study topological properties and enables operations such as the cup product, revealing deeper insights into the relationships between different spaces. Cohomology groups also exhibit properties like homotopy invariance and can be computed using various theories, including Alexandrov-Čech cohomology.
Cohomology operations: Cohomology operations are algebraic constructions that allow us to derive new cohomology classes from existing ones, providing a systematic way to explore the structure of cohomology rings. These operations help in understanding the relationships between different cohomology groups and can provide insights into the topology of spaces by linking their geometric properties to algebraic invariants.
Cohomology Ring: The cohomology ring is a mathematical structure that combines cohomology groups into a graded ring using the cup product operation. It encapsulates topological information about a space, allowing one to perform algebraic manipulations that reveal deeper insights into its geometric properties.
Cross Product: The cross product is a binary operation on two vectors in three-dimensional space, resulting in a new vector that is orthogonal to both of the original vectors. This operation is fundamental in various mathematical and physical contexts, as it helps in computing areas of parallelograms, determining torque, and analyzing rotations. Understanding the cross product is essential when working with cohomology operations, applying the Cartan formula, and exploring Pontryagin classes.
Cup product: The cup product is an operation in cohomology that combines two cohomology classes to produce a new cohomology class, allowing us to create a ring structure from the cohomology groups of a topological space. This operation plays a key role in understanding the algebraic properties of cohomology, connecting various concepts such as the cohomology ring, cohomology operations, and the Künneth formula.
De Rham cohomology: De Rham cohomology is a type of cohomology theory that uses differential forms to study the topology of smooth manifolds. It provides a powerful bridge between calculus and algebraic topology, allowing the study of manifold properties through the analysis of smooth functions and their derivatives.
Dimension: In mathematics, dimension refers to the minimum number of coordinates needed to specify a point within a given space. It connects to various concepts, such as the geometric structure of spaces and the algebraic properties of objects, which are essential for understanding relationships in areas like topology and algebraic geometry.
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.
Homotopy groups of spheres: Homotopy groups of spheres are algebraic structures that capture information about the higher-dimensional topology of spheres, denoted as $$\pi_n(S^m)$$, where $$n$$ and $$m$$ are non-negative integers representing dimensions. These groups are fundamental in algebraic topology, as they provide insight into the shape and structure of topological spaces through the lens of homotopy theory. They also play a crucial role in cohomology operations, allowing mathematicians to understand how various topological features interact and can be classified.
Künneth Formula: The Künneth Formula is a powerful result in algebraic topology that describes how the homology or cohomology groups of the product of two topological spaces relate to the homology or cohomology groups of the individual spaces. It provides a way to compute the homology or cohomology of a product space based on the known properties of its components, connecting directly to various aspects of algebraic topology, including operations and duality.
Naturality: Naturality refers to a property of mathematical structures that maintains their relationships and operations under a change of context or setting, showcasing how structures can be transformed while preserving essential characteristics. This concept is vital in various areas, highlighting the interplay between different mathematical objects and their homomorphisms, operations, and connecting mappings.
Obstruction Theory: Obstruction theory is a framework in algebraic topology that studies the conditions under which certain types of geometric or topological constructions can be achieved. It particularly focuses on the existence of sections and lifts, providing tools to determine when a desired structure can be realized in a specific setting. This concept plays a vital role in understanding the relationships between various cohomological constructs, impacting how we interpret cohomology groups, rings, and operations.
Singular Cohomology: Singular cohomology is a mathematical tool used in algebraic topology that assigns a sequence of abelian groups or vector spaces to a topological space, allowing us to study its global properties through the use of singular simplices. This concept connects the geometric aspects of spaces with algebraic structures, providing insights into various topological features such as holes and connectivity.
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 operations: Stable cohomology operations are algebraic constructions that act on the cohomology of topological spaces in a stable range, which typically refers to a situation where the spaces involved are sufficiently 'large' or 'complex' that their properties stabilize. These operations are crucial for understanding how cohomology behaves under various topological transformations and play a significant role in the development of stable homotopy theory, connecting algebraic topology with other mathematical fields.
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.
Topological invariants: Topological invariants are properties of a topological space that remain unchanged under continuous deformations, such as stretching or bending, but not tearing or gluing. These invariants help classify spaces and reveal essential features about their structure, playing a crucial role in various mathematical theories and applications.
Unstable cohomology operations: Unstable cohomology operations are algebraic functions that act on cohomology groups but do not respect the stability properties that arise in stable homotopy theory. They are particularly relevant in the study of cohomological phenomena in algebraic topology, as they provide insights into how different cohomology theories relate to one another and how they behave under various operations. These operations can lead to important results regarding the structure and relationships of spaces in terms of their cohomological dimensions and invariants.
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