2.2 Cohomology rings

Cohomology rings are powerful algebraic tools that capture the topological essence of spaces. By combining cohomology groups with the cup product, these structures offer a way to analyze global properties and multiplicative relationships between cohomology classes.

The study of cohomology rings involves exploring their definitions, examples, and properties. From spheres to projective spaces, these rings provide insights into various mathematical objects, while techniques like cellular cohomology and spectral sequences aid in their computation.

Definition of cohomology rings

• Cohomology rings are algebraic structures that capture intrinsic topological properties of spaces
• They provide a way to study the global structure of a space by associating a graded ring to it
• The ring structure is determined by the cup product, which allows for the multiplication of cohomology classes

Graded rings

Top images from around the web for 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 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

• A graded ring is a ring that decomposes into a direct sum of abelian groups indexed by integers, called degrees
• The multiplication in a graded ring respects the grading, meaning that the product of elements of degree $i$ and $j$ is an element of degree $i+j$
• Cohomology rings are examples of graded commutative rings, satisfying $ab = (-1)^{ij}ba$ for elements $a$ of degree $i$ and $b$ of degree $j$

Cup product

• The cup product is a bilinear operation that takes two cohomology classes and produces a new cohomology class of higher degree
• It is defined using the diagonal map of a space and the induced homomorphisms on cohomology
• The cup product is associative, graded commutative, and compatible with the coboundary operator

Ring structure

• The cohomology groups of a space, together with the cup product, form a graded ring
• The unit element is the cohomology class represented by the constant map to the coefficient ring
• The ring structure encodes information about the multiplicative properties of cohomology classes and their relationships

Examples of cohomology rings

Cohomology ring of spheres

• The cohomology ring of an $n$-sphere $S^n$ over a field $k$ is given by $H^*(S^n; k) = k[x]/(x^2)$, where $x$ is a generator of degree $n$
• For even-dimensional spheres, the cohomology ring is a truncated polynomial ring with a single generator
• For odd-dimensional spheres, the cohomology ring is an exterior algebra with a single generator

Cohomology ring of projective spaces

• The cohomology ring of the real projective space $\mathbb{RP}^n$ over $\mathbb{Z}/2\mathbb{Z}$ is given by $H^*(\mathbb{RP}^n; \mathbb{Z}/2\mathbb{Z}) = (\mathbb{Z}/2\mathbb{Z})[x]/(x^{n+1})$, where $x$ is a generator of degree $1$
• The cohomology ring of the complex projective space $\mathbb{CP}^n$ over $\mathbb{Z}$ is given by $H^*(\mathbb{CP}^n; \mathbb{Z}) = \mathbb{Z}[x]/(x^{n+1})$, where $x$ is a generator of degree $2$

Cohomology ring of surfaces

• The cohomology ring of a closed orientable surface $\Sigma_g$ of genus $g$ over a field $k$ is given by $H^*(\Sigma_g; k) = k[a, b]/(a^2, b^2, ab - ba)$, where $a$ and $b$ are generators of degree $1$
• The cohomology ring of a closed non-orientable surface $N_h$ with $h$ crosscaps over $\mathbb{Z}/2\mathbb{Z}$ is given by $H^*(N_h; \mathbb{Z}/2\mathbb{Z}) = (\mathbb{Z}/2\mathbb{Z})[x, y]/(x^2 + xy + y^2, x^h)$, where $x$ and $y$ are generators of degree $1$

Properties of cohomology rings

Functoriality

• Cohomology rings are functorial, meaning that continuous maps between spaces induce homomorphisms between their cohomology rings
• The induced homomorphisms are compatible with the cup product and preserve the graded ring structure
• Functoriality allows for the study of maps between spaces through their effect on cohomology rings

Künneth formula

• The Künneth formula relates the cohomology ring of a product space to the cohomology rings of its factors
• For spaces $X$ and $Y$, there is an isomorphism of graded rings $H^*(X \times Y; R) \cong H^*(X; R) \otimes_R H^*(Y; R)$, where $R$ is the coefficient ring
• The isomorphism is given by the cross product, which is compatible with the cup product on each factor

Poincaré duality

• Poincaré duality establishes a relationship between the cohomology and homology of a closed orientable manifold
• For a closed orientable $n$-manifold $M$, there is an isomorphism $H^k(M; R) \cong H_{n-k}(M; R)$ for any coefficient ring $R$
• The cup product in cohomology corresponds to the intersection product in homology under this isomorphism

Computation of cohomology rings

Cellular cohomology

• Cellular cohomology is a method for computing the cohomology ring of a CW complex using its cellular structure
• Each cell of the complex contributes a generator to the cellular cochain complex, and the coboundary maps are determined by the attaching maps
• The cellular cohomology ring is isomorphic to the singular cohomology ring and can be computed using linear algebra

Spectral sequences

• Spectral sequences are algebraic tools that provide a systematic way to compute cohomology rings in certain situations
• They are derived from filtrations or double complexes and involve a sequence of pages, each containing a bigraded module with differentials
• The Serre spectral sequence and the Leray-Serre spectral sequence are examples used to compute cohomology rings of fiber bundles and fibrations

Characteristic classes

• Characteristic classes are cohomology classes associated with vector bundles or principal bundles over a space
• They provide obstructions to the existence of certain structures on the bundles, such as orientations or trivializations
• Examples include Stiefel-Whitney classes, Chern classes, and Pontryagin classes, which live in the cohomology rings of the base space

Applications of cohomology rings

Intersection theory

• Intersection theory studies the intersection of submanifolds or cycles in a manifold using cohomological methods
• The cup product in cohomology corresponds to the intersection of submanifolds under Poincaré duality
• Cohomology rings provide a framework for computing intersection numbers and understanding the geometric properties of intersections

Obstruction theory

• Obstruction theory uses cohomology to study the existence and classification of certain geometric structures, such as cross-sections of bundles or lifts of maps
• Cohomology classes serve as obstructions to the existence of these structures, and their vanishing is often a necessary and sufficient condition
• The obstruction classes live in the cohomology rings of the spaces involved and can be computed using spectral sequences or characteristic classes

Steenrod operations

• Steenrod operations are additional structures on the cohomology rings of spaces, providing a finer invariant than the cup product alone
• They are natural transformations between cohomology functors that satisfy certain properties, such as the Cartan formula and the Adem relations
• Steenrod operations can be used to distinguish between spaces with isomorphic cohomology rings and to study the homotopy theory of spaces