5.3 Cohomology rings

Cohomology rings are powerful algebraic structures that encode topological information about spaces. They combine cohomology groups with the cup product 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

Top images from around the web for Cohomology groups as graded rings

• 

  L∞-algebras and their cohomology | Emergent Scientist View original

  Is this image relevant?

• 

  L∞-algebras and their cohomology | Emergent Scientist View original

  Is this image relevant?

• 

  L∞-algebras and their cohomology | Emergent Scientist View original

  Is this image relevant?

• 

  L∞-algebras and their cohomology | Emergent Scientist View original

  Is this image relevant?

• 

  L∞-algebras and their cohomology | Emergent Scientist View original

  Is this image relevant?

1 of 3

Top images from around the web for Cohomology groups as graded rings

• 

  L∞-algebras and their cohomology | Emergent Scientist View original

  Is this image relevant?

• 

  L∞-algebras and their cohomology | Emergent Scientist View original

  Is this image relevant?

• 

  L∞-algebras and their cohomology | Emergent Scientist View original

  Is this image relevant?

• 

  L∞-algebras and their cohomology | Emergent Scientist View original

  Is this image relevant?

• 

  L∞-algebras and their cohomology | Emergent Scientist View original

  Is this image relevant?

1 of 3

• Cohomology groups $H^n(X; R)$ form a graded abelian group over a ring $R$ for a topological space $X$
• The grading is given by the cohomological degree $n$, with $H^n(X; R)$ representing the $n$-th cohomology group
• The direct sum of cohomology groups $\bigoplus_{n \geq 0} H^n(X; R)$ has a natural graded ring structure induced by the cup product

Cup product operation

• The cup product is a bilinear operation $\smile: H^p(X; R) \times H^q(X; R) \to H^{p+q}(X; R)$ that combines cohomology classes of degrees $p$ and $q$ to produce a cohomology class of degree $p+q$
• Defined using the diagonal map $\Delta: X \to X \times X$ 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 cohomology ring satisfies the axioms of a graded commutative ring
• The cup product is associative: $(\alpha \smile \beta) \smile \gamma = \alpha \smile (\beta \smile \gamma)$ for cohomology classes $\alpha, \beta, \gamma$
• The cup product distributes over addition: $\alpha \smile (\beta + \gamma) = \alpha \smile \beta + \alpha \smile \gamma$ and $(\alpha + \beta) \smile \gamma = \alpha \smile \gamma + \beta \smile \gamma$
• The identity element is the class $1 \in H^0(X; R)$, satisfying $1 \smile \alpha = \alpha \smile 1 = \alpha$ for any cohomology class $\alpha$

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 $S^n$ over a field $k$ is given by $H^*(S^n; k) \cong k[x]/(x^2)$, where $x$ is a generator of degree $n$
• The ring structure is determined by the fact that $x \smile x = 0$ since the cup product of the generator with itself vanishes
• For example, the cohomology ring of the circle $S^1$ is $H^*(S^1; k) \cong k[x]/(x^2)$ with $|x| = 1$, and the cohomology ring of the 2-sphere $S^2$ is $H^*(S^2; k) \cong k[y]/(y^2)$ with $|y| = 2$

Cohomology ring of projective spaces

• The cohomology ring of the real projective space $\mathbb{R}P^n$ over $\mathbb{Z}/2\mathbb{Z}$ is $H^*(\mathbb{R}P^n; \mathbb{Z}/2\mathbb{Z}) \cong (\mathbb{Z}/2\mathbb{Z})[x]/(x^{n+1})$, where $x$ is a generator of degree 1
• The cohomology ring of the complex projective space $\mathbb{C}P^n$ over $\mathbb{Z}$ is $H^*(\mathbb{C}P^n; \mathbb{Z}) \cong \mathbb{Z}[y]/(y^{n+1})$, 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 $\Sigma_g$ of genus $g$ over a field $k$ is $H^*(\Sigma_g; k) \cong k[x, y]/(x^2, y^2, xy - yx)$, where $x$ and $y$ are generators of degree 1
• The relations $x^2 = y^2 = 0$ 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 $T^2$ is $H^*(T^2; k) \cong k[x, y]/(x^2, y^2, xy - yx)$ 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 $\alpha \in H^p(X; R)$ and $\beta \in H^q(X; R)$, we have $\alpha \smile \beta = (-1)^{pq} \beta \smile \alpha$
• This property arises from the sign convention in the definition of the cup product and the commutativity of the coefficient ring $R$
• Graded commutativity simplifies computations and provides a symmetry in the structure of the cohomology ring

Cohomology ring homomorphisms

• Continuous maps $f: X \to Y$ induce homomorphisms of cohomology rings $f^*: H^*(Y; R) \to H^*(X; R)$ that preserve the cup product structure
• The induced homomorphisms are compatible with the grading and satisfy $f^*(\alpha \smile \beta) = f^*(\alpha) \smile f^*(\beta)$ for cohomology classes $\alpha, \beta \in H^*(Y; R)$
• Cohomology ring homomorphisms allow for the comparison and transfer of cohomological information between spaces

Künneth formula for cohomology rings

• The Künneth formula describes the cohomology ring of a product space $X \times Y$ in terms of the cohomology rings of the factors $X$ and $Y$
• Under suitable conditions, there is an isomorphism of graded rings $H^*(X \times Y; R) \cong H^*(X; R) \otimes_R H^*(Y; R)$, where $\otimes_R$ denotes the tensor product over the coefficient ring $R$
• The cross product of cohomology classes $\alpha \in H^p(X; R)$ and $\beta \in H^q(Y; R)$ is defined as $\alpha \times \beta = \pi_X^*(\alpha) \smile \pi_Y^*(\beta) \in H^{p+q}(X \times Y; R)$, where $\pi_X$ and $\pi_Y$ 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

• 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 $Sq^i: H^n(X; \mathbb{Z}/2\mathbb{Z}) \to H^{n+i}(X; \mathbb{Z}/2\mathbb{Z})$ and the Steenrod reduced powers $P^i: H^n(X; \mathbb{Z}/p\mathbb{Z}) \to H^{n+2i(p-1)}(X; \mathbb{Z}/p\mathbb{Z})$ 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 $X \vee Y$ is related to the cohomology rings of the individual spaces $X$ and $Y$
• For pointed spaces, there is a split short exact sequence of graded rings $0 \to \tilde{H}^*(X; R) \oplus \tilde{H}^*(Y; R) \to H^*(X \vee Y; R) \to R \to 0$, where $\tilde{H}^*$ denotes the reduced cohomology
• The cup product in $H^*(X \vee Y; R)$ 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 $X \times Y$ 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 $H^*(X \times Y; R) \cong H^*(X; R) \otimes_R H^*(Y; R)$
• 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 $S^n \to E \to B$, there is a long exact sequence of cohomology rings $\cdots \to H^{k-n}(B; R) \xrightarrow{\smile e} H^k(B; R) \to H^k(E; R) \to H^{k-n+1}(B; R) \to \cdots$, where $e \in H^n(B; R)$ 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 singular cohomology
• 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 cochain complex 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 $H^k_{dR}(M; \mathbb{R})$ 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 $\check{H}^k(X; R)$ 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