of spaces is a powerful tool in algebraic topology, assigning abelian groups to topological spaces to capture their structure. It's the dual of homology, focusing on functions called cochains instead of chains, and provides insights into the "holes" in spaces.
This topic explores key concepts like cochains, cocycles, and coboundaries, as well as computational tools like the . It also covers , manifold cohomology, , and connections to homotopy theory, highlighting cohomology's wide-ranging applications in mathematics.
Definition of cohomology
Cohomology is a powerful algebraic tool used to study the global properties of topological spaces
Assigns abelian groups () to a topological space, capturing essential information about its structure
Dual notion to homology, focusing on functions (cochains) rather than chains
Cochains vs cocycles
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Cochains are functions that assign abelian group elements to simplices of a given dimension
Cocycles are cochains that satisfy a specific condition: their coboundary is zero
Cocycles represent closed forms and capture global properties of the space
Cochains that are not cocycles (coboundaries) correspond to exact forms and can be "divided out"
Coboundaries and quotients
Coboundaries are cochains that are the coboundary of a lower-dimensional cochain
The set of coboundaries forms a subgroup of the group of cochains
The cohomology group is defined as the quotient of the group of cocycles by the group of coboundaries
This quotient construction ensures that cohomology is independent of the choice of representative cocycles
Cohomology groups
Cohomology groups Hn(X;G) are associated with a topological space X and an abelian group G
The degree n represents the dimension of the cochains being considered
Elements of the cohomology group are equivalence classes of cocycles (cocycles that differ by a coboundary are considered equivalent)
Cohomology groups provide a measure of the "holes" or "obstructions" in the space X
The rank of the cohomology group (Betti number) counts the number of independent holes in each dimension
Computational tools
Several powerful computational tools are available to calculate cohomology groups in various settings
These tools allow for the computation of cohomology in more complex spaces by breaking them down into simpler pieces
Computational tools often involve long exact sequences or spectral sequences
Mayer-Vietoris sequence
Relates the cohomology of a space X to the cohomology of two overlapping subspaces A and B
Provides a long involving the cohomology groups of X, A, B, and their intersection A∩B
Allows for the computation of cohomology by decomposing a space into simpler parts
Particularly useful when the cohomology of the subspaces and their intersection is known or easier to compute
Künneth formula
Computes the cohomology of a product space X×Y in terms of the cohomology of the individual spaces X and Y
Expresses the cohomology groups of the product as a tensor product of the cohomology groups of the factors
Provides a way to calculate the cohomology of higher-dimensional spaces built from lower-dimensional ones
Requires certain conditions on the coefficient ring (e.g., a field) for the formula to hold
Universal coefficient theorem
Relates the cohomology of a space X with coefficients in different abelian groups
Establishes a relationship between cohomology with integer coefficients and cohomology with coefficients in any abelian group G
Involves the Ext functor, which captures torsion information
Allows for the computation of cohomology with various coefficients based on the integral cohomology
Cohomology operations
Cohomology operations are additional structures on cohomology groups that provide further insight into the topological space
These operations allow for the combination and manipulation of cohomology classes
Cohomology operations often have geometric interpretations and connections to other invariants
Cup product
A multiplicative operation on cohomology classes, generalizing the wedge product of differential forms
Takes two cohomology classes α∈Hp(X;R) and β∈Hq(X;R) and produces a new class α⌣β∈Hp+q(X;R)
Satisfies graded commutativity: α⌣β=(−1)pqβ⌣α
Provides a ring structure on the direct sum of cohomology groups ⨁nHn(X;R)
Allows for the study of the multiplicative structure of cohomology and its relation to the geometry of the space
Cap product and Poincaré duality
The is a bilinear pairing between cohomology and homology classes
Takes a cohomology class α∈Hp(X;R) and a homology class σ∈Hq(X;R) and produces a new homology class α⌢σ∈Hq−p(X;R)
relates the cohomology of a compact oriented manifold M to its homology via the cap product
For a compact oriented n-manifold M, the map D:Hk(M;R)→Hn−k(M;R) given by D(α)=α⌢[M] is an isomorphism
Poincaré duality allows for the translation of cohomological statements into homological ones and vice versa
Cohomology of manifolds
The has special properties and connections to other geometric structures
Smooth manifolds admit a natural cohomology theory called , based on differential forms
The study of cohomology on manifolds leads to powerful results and applications in geometry and physics
De Rham cohomology
De Rham cohomology is a cohomology theory for smooth manifolds based on differential forms
The cochains are differential forms of various degrees, with the exterior derivative d serving as the coboundary operator
The k-th de Rham cohomology group HdRk(M) is the quotient of closed k-forms by exact k-forms
De Rham's theorem establishes an isomorphism between de Rham cohomology and with real coefficients
Provides a link between the algebraic world of cohomology and the world of differential forms and calculus on manifolds
Hodge theory and harmonic forms
studies the relationship between the de Rham cohomology of a compact Riemannian manifold and the space of
Harmonic forms are differential forms that are both closed (dω=0) and co-closed (d∗ω=0), where d∗ is the adjoint of the exterior derivative
The Hodge decomposition theorem states that every differential form can be uniquely written as the sum of a harmonic form, an exact form, and a co-exact form
The space of harmonic k-forms is isomorphic to the k-th de Rham cohomology group
Hodge theory provides a powerful tool for studying the topology of manifolds using geometric and analytic techniques
Characteristic classes
Characteristic classes are cohomology classes associated with vector bundles or principal bundles over a topological space
They provide a way to measure the "twisting" or non-triviality of the bundle
Different types of characteristic classes exist for different kinds of bundles and have various geometric and topological interpretations
Chern classes
are characteristic classes associated with complex vector bundles
The k-th Chern class ck(E) of a complex vector bundle E is an element of the 2k-th cohomology group of the base space
Chern classes satisfy certain axioms, including naturality and the Whitney product formula
The total Chern class c(E)=1+c1(E)+c2(E)+⋯ encodes all the Chern classes of a bundle
Chern classes have important applications in algebraic geometry, complex geometry, and topology
Pontryagin classes
are characteristic classes associated with real vector bundles
The k-th Pontryagin class pk(E) of a real vector bundle E is an element of the 4k-th cohomology group of the base space
Pontryagin classes are defined using the Chern classes of the complexification of the real vector bundle
They provide topological obstructions to the existence of certain geometric structures on manifolds (e.g., almost complex structures)
Pontryagin classes are related to the signature of a manifold and have applications in differential topology and geometry
Stiefel-Whitney classes
are characteristic classes associated with real vector bundles, taking values in mod 2 cohomology
The k-th Stiefel-Whitney class wk(E) of a real vector bundle E is an element of the k-th mod 2 cohomology group of the base space
Stiefel-Whitney classes satisfy certain axioms, including naturality and the Whitney product formula
The total Stiefel-Whitney class w(E)=1+w1(E)+w2(E)+⋯ encodes all the Stiefel-Whitney classes of a bundle
Stiefel-Whitney classes provide information about the orientability and the existence of certain structures on vector bundles and manifolds
Cohomology and homotopy theory
Cohomology and homotopy theory are closely related, with cohomology providing algebraic invariants of homotopy types
and are key constructions that link cohomology and homotopy
uses cohomology to study the existence and uniqueness of certain maps between spaces
Cohomology of Eilenberg-MacLane spaces
Eilenberg-MacLane spaces K(G,n) are spaces with a single non-trivial homotopy group πn(K(G,n))≅G
The cohomology of Eilenberg-MacLane spaces is given by the cohomology of the group G: H∗(K(G,n);A)≅H∗(G;A) for any abelian group A
This isomorphism allows for the interpretation of group cohomology in terms of the cohomology of a topological space
The cohomology of Eilenberg-MacLane spaces plays a crucial role in the classification of homotopy types and the construction of Postnikov towers
Postnikov towers and obstruction theory
A Postnikov tower is a sequence of spaces Xn that approximate a given space X, with each Xn capturing the homotopy groups of X up to dimension n
The successive stages of the Postnikov tower are related by fibrations with fibers being Eilenberg-MacLane spaces
The cohomology groups of X can be studied using the cohomology of the stages of its Postnikov tower
Obstruction theory uses cohomology to determine the existence and uniqueness of lifts in the Postnikov tower
The obstructions to lifting a map f:Y→Xn to a map f~:Y→Xn+1 lie in certain cohomology groups of Y with coefficients in the homotopy groups of the fiber
Applications and examples
Cohomology has numerous applications in various branches of mathematics, including algebraic topology, differential geometry, and algebraic geometry
Many geometric and topological invariants can be expressed in terms of cohomology classes and their properties
Examples include intersection theory, degree theory, and the study of characteristic classes
Intersection theory
Intersection theory studies the intersection of submanifolds or cycles in a manifold
The in cohomology provides a way to define the intersection product of cycles
The intersection of two submanifolds can be computed by evaluating the cup product of their Poincaré dual cohomology classes
Intersection theory has applications in enumerative geometry, counting the number of solutions to geometric problems
Degree and winding number
The degree of a continuous map f:Sn→Sn is an integer that measures the "wrapping" of the domain sphere around the codomain sphere
The degree can be computed using the induced map on top cohomology f∗:Hn(Sn)→Hn(Sn), which is multiplication by the degree
The winding number of a curve in the plane around a point is a special case of the degree, measuring how many times the curve winds around the point
Degree theory has applications in fixed point theory, nonlinear analysis, and the study of critical points of functions
Euler class and Euler characteristic
The Euler class is a characteristic class associated with oriented vector bundles
For an oriented rank n vector bundle E over a space X, the Euler class e(E) is an element of the n-th cohomology group Hn(X;Z)
The Euler class measures the obstruction to the existence of a non-vanishing section of the vector bundle
The Euler characteristic of a compact oriented manifold M can be expressed as the evaluation of the Euler class of the tangent bundle on the fundamental class: χ(M)=⟨e(TM),[M]⟩
The Euler characteristic is a topological invariant that provides information about the shape and structure of the manifold
Key Terms to Review (23)
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.
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.
Chern classes: Chern classes are topological invariants associated with complex vector bundles that provide crucial information about the geometry and topology of the underlying space. They capture characteristics like curvature and the way bundles twist and turn, connecting deeply with other concepts like cohomology, characteristic classes, and various forms of K-theory.
Cohomology: Cohomology is a mathematical tool in algebraic topology that assigns a sequence of abelian groups or vector spaces to a topological space, capturing information about its shape and structure. This concept helps to analyze spaces by providing invariants that are useful for distinguishing between them and studying their properties through cochains, cocycles, and cohomology classes. Cohomology can be applied to various contexts such as spaces, pairs of spaces, and even spectral sequences, revealing deep connections among different areas of mathematics.
Cohomology Groups: Cohomology groups are algebraic structures that assign a sequence of abelian groups or modules to a topological space, providing a way to classify the space's shape and features. These groups arise from the study of cochains, which are functions defined on the simplices of a given space, allowing for insights into the structure and properties of both spaces and groups through their interactions.
Cohomology of Manifolds: Cohomology of manifolds refers to a mathematical framework used to study the properties of manifolds through cohomological techniques. It provides a way to associate algebraic invariants to topological spaces, revealing information about their structure, such as holes and other features. This concept connects deeply with cohomology of spaces and the Alexandrov-Čech cohomology, allowing for various approaches in understanding how manifolds behave under continuous transformations.
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.
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.
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.
Exact Sequence: An exact sequence is a sequence of algebraic objects and morphisms between them where the image of one morphism is equal to the kernel of the next. This concept is crucial in connecting different algebraic structures, and it plays an essential role in understanding relationships between homology and cohomology groups, providing a powerful tool for studying topological spaces.
Harmonic Forms: Harmonic forms are differential forms on a Riemannian manifold that are both closed and co-closed, meaning they satisfy specific conditions related to the Laplace operator. These forms play a crucial role in understanding the cohomology of spaces, particularly when analyzing the relationship between geometry and topology. They provide insight into how differential forms behave under the influence of the manifold's structure, leading to significant results in various areas of mathematics, including Hodge theory.
Hodge Theory: Hodge Theory is a powerful mathematical framework that connects algebraic topology and differential geometry through the study of harmonic forms on a manifold. It reveals how the topology of a space relates to the analysis of differential forms, establishing key relationships between cohomology groups and the space of harmonic forms. This connection is pivotal for understanding various forms of cohomology, including de Rham cohomology, and has significant implications for many areas in mathematics.
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.
Mayer-Vietoris sequence: The Mayer-Vietoris sequence is a powerful tool in algebraic topology that provides a way to compute the homology and cohomology groups of a topological space by decomposing it into simpler pieces. It connects the homology and cohomology of two overlapping subspaces with that of their union, forming a long exact sequence that highlights the relationships between these spaces.
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.
Poincaré Duality: Poincaré Duality is a fundamental theorem in algebraic topology that establishes a relationship between the cohomology groups of a manifold and its homology groups, particularly in the context of closed oriented manifolds. This duality implies that the k-th cohomology group of a manifold is isomorphic to the (n-k)-th homology group, where n is the dimension of the manifold, revealing deep connections between these two areas of topology.
Pontryagin classes: Pontryagin classes are topological invariants associated with real vector bundles, providing important information about the curvature of the bundles. They arise from the Chern-Weil theory and are related to the characteristic classes of a manifold, capturing geometric and topological properties. These classes play a crucial role in the study of the cohomology of spaces and in the formulation of Wu classes, linking algebraic topology with differential geometry.
Postnikov Towers: Postnikov towers are a method used in algebraic topology to break down a space into simpler pieces that can be more easily studied, specifically by utilizing fibrations and Eilenberg-MacLane spaces. Each stage of a Postnikov tower captures certain cohomological information about the space, which helps in understanding its overall structure and properties through successive approximations. This concept is particularly useful for analyzing the cohomology of spaces, allowing us to connect homotopy theory with cohomological techniques.
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 Sequence: A spectral sequence is a mathematical tool used in algebraic topology and homological algebra that provides a systematic way to compute homology or cohomology groups by organizing data into a sequence of pages, each with its own differential structure. This concept allows one to break down complex calculations into more manageable pieces, revealing relationships between different cohomology groups and simplifying the analysis of topological spaces or spectra.
Stiefel-Whitney classes: Stiefel-Whitney classes are characteristic classes associated with real vector bundles, providing important topological invariants that help classify these bundles. They play a significant role in the study of manifold properties, particularly in relation to cohomology theories, where they reveal information about the intersection of submanifolds and the topology of vector bundles. These classes are particularly useful for understanding orientability and the existence of certain structures on manifolds.
Universal Coefficient Theorem: The Universal Coefficient Theorem provides a relationship between homology and cohomology groups, allowing the computation of cohomology groups based on homology groups and Ext and Tor functors. It serves as a bridge between algebraic topology and homological algebra, illustrating how these concepts interact across various mathematical contexts.