Cohomology rings are powerful algebraic structures that encode topological information about spaces. They combine cohomology groups with the operation, providing a graded ring structure that captures essential properties of topological spaces.
These rings are fundamental tools in algebraic topology, used to study and classify spaces. By examining the structure and properties of cohomology rings, we can gain insights into the underlying geometry and relationships between different topological spaces.
Definition of cohomology rings
Cohomology rings provide a powerful algebraic structure that encodes topological information about spaces
Combines the graded abelian group structure of cohomology with a multiplicative operation called the cup product
Cohomology rings are fundamental tools in algebraic topology for studying the properties and relationships between topological spaces
Cohomology groups as graded rings
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Cohomology groups Hn(X;R) form a graded abelian group over a ring R for a topological space X
The grading is given by the cohomological degree n, with Hn(X;R) representing the n-th cohomology group
The direct sum of cohomology groups ⨁n≥0Hn(X;R) has a natural graded ring structure induced by the cup product
Cup product operation
The cup product is a bilinear operation ⌣:Hp(X;R)×Hq(X;R)→Hp+q(X;R) that combines cohomology classes of degrees p and q to produce a cohomology class of degree p+q
Defined using the diagonal map Δ:X→X×X and the induced homomorphisms on cohomology
The cup product is associative and distributes over addition, making the cohomology groups a graded ring
Ring structure axioms
The satisfies the axioms of a graded commutative ring
The cup product is associative: (α⌣β)⌣γ=α⌣(β⌣γ) for cohomology classes α,β,γ
The cup product distributes over addition: α⌣(β+γ)=α⌣β+α⌣γ and (α+β)⌣γ=α⌣γ+β⌣γ
The identity element is the class 1∈H0(X;R), satisfying 1⌣α=α⌣1=α for any cohomology class α
Examples of cohomology rings
Cohomology rings provide a rich source of invariants for distinguishing and classifying topological spaces
Studying the structure and properties of cohomology rings helps understand the underlying geometry and topology of spaces
Explicit computations of cohomology rings for specific spaces reveal their intrinsic characteristics and relationships
Cohomology ring of spheres
The cohomology ring of the n-sphere Sn over a field k is given by H∗(Sn;k)≅k[x]/(x2), where x is a generator of degree n
The ring structure is determined by the fact that x⌣x=0 since the cup product of the generator with itself vanishes
For example, the cohomology ring of the circle S1 is H∗(S1;k)≅k[x]/(x2) with ∣x∣=1, and the cohomology ring of the 2-sphere S2 is H∗(S2;k)≅k[y]/(y2) with ∣y∣=2
Cohomology ring of projective spaces
The cohomology ring of the real projective space RPn over Z/2Z is H∗(RPn;Z/2Z)≅(Z/2Z)[x]/(xn+1), where x is a generator of degree 1
The cohomology ring of the complex projective space CPn over Z is H∗(CPn;Z)≅Z[y]/(yn+1), where y is a generator of degree 2
These cohomology rings capture the distinctive properties of projective spaces and their underlying field of scalars
Cohomology ring of surfaces
The cohomology ring of a closed orientable surface Σg of genus g over a field k is H∗(Σg;k)≅k[x,y]/(x2,y2,xy−yx), where x and y are generators of degree 1
The relations x2=y2=0 and xy=yx reflect the cup product structure and the fact that the square of any 1-dimensional class vanishes on a surface
For example, the cohomology ring of the torus T2 is H∗(T2;k)≅k[x,y]/(x2,y2,xy−yx) with ∣x∣=∣y∣=1, capturing its essential topological features
Properties of cohomology rings
Cohomology rings exhibit several important properties that reflect the underlying topological and algebraic structures
These properties provide insights into the behavior of cohomology classes under various operations and transformations
Understanding the properties of cohomology rings is crucial for computing and manipulating cohomology in applications
Graded commutativity
The cohomology ring is graded commutative, meaning that for cohomology classes α∈Hp(X;R) and β∈Hq(X;R), we have α⌣β=(−1)pqβ⌣α
This property arises from the sign convention in the definition of the cup product and the commutativity of the coefficient ring R
simplifies computations and provides a symmetry in the structure of the cohomology ring
Cohomology ring homomorphisms
Continuous maps f:X→Y induce homomorphisms of cohomology rings f∗:H∗(Y;R)→H∗(X;R) that preserve the cup product structure
The induced homomorphisms are compatible with the grading and satisfy f∗(α⌣β)=f∗(α)⌣f∗(β) for cohomology classes α,β∈H∗(Y;R)
Cohomology ring homomorphisms allow for the comparison and transfer of cohomological information between spaces
Künneth formula for cohomology rings
The describes the cohomology ring of a product space X×Y in terms of the cohomology rings of the factors X and Y
Under suitable conditions, there is an of graded rings H∗(X×Y;R)≅H∗(X;R)⊗RH∗(Y;R), where ⊗R denotes the tensor product over the coefficient ring R
The cross product of cohomology classes α∈Hp(X;R) and β∈Hq(Y;R) is defined as α×β=πX∗(α)⌣πY∗(β)∈Hp+q(X×Y;R), where πX and πY are the projections onto X and Y, respectively
Applications of cohomology rings
Cohomology rings have numerous applications in algebraic topology, geometry, and related fields
They provide powerful tools for studying and classifying topological spaces, vector bundles, and other geometric structures
Cohomology rings also play a crucial role in obstruction theory and the computation of
Characteristic classes
Characteristic classes are cohomology classes associated with vector bundles that capture their topological and geometric properties
The Chern classes of a complex vector bundle and the Stiefel-Whitney classes of a real vector bundle are examples of characteristic classes that live in the cohomology rings of the base space
Characteristic classes provide obstructions to the existence of certain geometric structures and are used to classify vector bundles up to isomorphism
Obstruction theory
Obstruction theory uses cohomology rings to study the existence and uniqueness of continuous maps between spaces satisfying certain conditions
The obstructions to extending maps or constructing sections of fibrations are often expressed as cohomology classes in specific degrees
Cohomology rings provide a framework for computing and analyzing these obstructions, leading to important results in homotopy theory and fiber bundle theory
Steenrod operations on cohomology rings
Steenrod operations are a family of cohomology operations that act on the cohomology rings of spaces, preserving the cup product structure
The Steenrod squares Sqi:Hn(X;Z/2Z)→Hn+i(X;Z/2Z) and the Steenrod reduced powers Pi:Hn(X;Z/pZ)→Hn+2i(p−1)(X;Z/pZ) are examples of Steenrod operations for p=2 and odd primes p, respectively
Steenrod operations provide additional structure on cohomology rings and are used to derive cohomological invariants and obstructions
Computations with cohomology rings
Computing the cohomology rings of spaces is a central problem in algebraic topology
Various techniques and tools are employed to determine the structure and generators of cohomology rings
Computations often involve the use of long exact sequences, spectral sequences, and other algebraic machinery
Cohomology ring of wedge sums
The cohomology ring of a wedge sum of spaces X∨Y is related to the cohomology rings of the individual spaces X and Y
For pointed spaces, there is a split short of graded rings 0→H~∗(X;R)⊕H~∗(Y;R)→H∗(X∨Y;R)→R→0, where H~∗ denotes the reduced cohomology
The cup product in H∗(X∨Y;R) is determined by the cup products in the cohomology rings of X and Y and the splitting of the sequence
Cohomology ring of product spaces
The cohomology ring of a product space X×Y is related to the cohomology rings of the factors X and Y via the Künneth formula
Under suitable conditions, there is an isomorphism of graded rings H∗(X×Y;R)≅H∗(X;R)⊗RH∗(Y;R)
The cross product of cohomology classes provides a means to compute the cup product in the cohomology ring of the product space
Gysin sequence and cohomology rings
The Gysin sequence is a long exact sequence that relates the cohomology rings of a sphere bundle and its base space
For an oriented sphere bundle Sn→E→B, there is a long exact sequence of cohomology rings ⋯→Hk−n(B;R)⌣eHk(B;R)→Hk(E;R)→Hk−n+1(B;R)→⋯, where e∈Hn(B;R) is the Euler class of the bundle
The Gysin sequence provides a powerful tool for computing the cohomology rings of sphere bundles and related spaces
Relation to other cohomology theories
Cohomology rings can be defined and studied in various cohomology theories beyond
Different cohomology theories offer alternative perspectives and computational techniques for understanding the cohomological structure of spaces
Comparing and relating cohomology rings across different theories provides a comprehensive view of the topological and geometric information they capture
Singular vs. simplicial cohomology rings
Singular cohomology and simplicial cohomology are two different approaches to defining cohomology groups and rings
Singular cohomology is based on the singular chain complex, while simplicial cohomology uses the simplicial of a simplicial complex
For simplicial complexes, the singular and simplicial cohomology rings are naturally isomorphic, providing a bridge between the two theories
De Rham cohomology ring
De Rham cohomology is a cohomology theory defined for smooth manifolds using differential forms
The De Rham cohomology groups HdRk(M;R) of a smooth manifold M are the cohomology groups of the complex of differential forms with the exterior derivative
The wedge product of differential forms induces a cup product on De Rham cohomology, making it a graded ring isomorphic to the singular cohomology ring with real coefficients
Čech cohomology ring
Čech cohomology is a cohomology theory defined using open covers of a topological space
The Čech cohomology groups Hˇk(X;R) are the direct limits of the cohomology groups of the nerve complexes associated with open covers of X
The cup product in Čech cohomology is defined using the refinement of open covers and the induced homomorphisms on cohomology
For sufficiently nice spaces, such as manifolds or CW complexes, the Čech cohomology ring is isomorphic to the singular cohomology ring