7.2 Cohomology rings

Cohomology rings take cohomology groups to the next level. They use the cup product to multiply cohomology classes, revealing deeper connections between different degrees. This structure helps us see topological features that individual groups might miss.

Cohomology rings are powerful tools for telling spaces apart. Even when two spaces have the same cohomology groups, their rings might be different. This extra info helps us spot subtle differences and understand spaces better than homology alone can.

Cohomology ring of a space

Definition and properties

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• The cohomology ring of a topological space X is a graded ring, denoted H*(X), with the cup product as the multiplication operation
• The cup product is a bilinear map ⌣: Hᵖ(X) × Hᵗ(X) → Hᵖ⁺ᵗ(X) takes two cohomology classes and produces a new cohomology class in a higher degree
  • The cup product is associative, meaning $(a ⌣ b) ⌣ c = a ⌣ (b ⌣ c)$ for cohomology classes a, b, and c
  • The cup product is distributive over addition, meaning $a ⌣ (b + c) = (a ⌣ b) + (a ⌣ c)$ for cohomology classes a, b, and c
• The cohomology ring encodes important topological information about the space X
  • Multiplicative structure reveals relationships between cohomology classes in different degrees
  • Ring structure can detect higher-order topological features not apparent from individual cohomology groups

Significance and applications

• The cohomology ring is a powerful invariant of a topological space
  • Spaces with non-isomorphic cohomology rings cannot be homotopy equivalent
  • Cohomology ring can distinguish spaces with isomorphic cohomology groups (ℂP² and S² × S⁴)
• Cohomology ring computations provide insight into the structure of a space
  • Generators and relations of the ring reveal essential topological features
  • Ring structure can be used to compute other invariants (characteristic classes, Massey products)
• Applications in algebraic topology and related fields
  • Cohomology rings are used to study vector bundles, fiber bundles, and principal G-bundles
  • Cohomology rings play a central role in obstruction theory and the classification of manifolds
  • Cohomology ring computations are essential in algebraic geometry and representation theory

Cohomology ring computations

Basic spaces and techniques

• The cohomology ring of the n-sphere, H*(Sⁿ), is isomorphic to ℤ[α]/〈α²〉
  • α is a generator of degree n
  • 〈α²〉 denotes the ideal generated by α², meaning α ⌣ α = 0
• The cohomology ring of real projective n-space, H*(ℝPⁿ), is isomorphic to ℤ[α]/〈2α,α^(n+1)〉
  • α is a generator of degree 1
  • Relations 2α = 0 and α^(n+1) = 0 hold in the ring
• The cohomology ring of complex projective n-space, H*(ℂPⁿ), is isomorphic to ℤ[α]/〈α^(n+1)〉
  • α is a generator of degree 2
  • Relation α^(n+1) = 0 holds in the ring
• To compute the cohomology ring of a product space X × Y, use the Künneth formula
  • H*(X × Y) ≅ H*(X) ⊗ H*(Y) as graded rings
  • The cup product in H*(X × Y) is determined by $(a ⊗ b) ⌣ (c ⊗ d) = (a ⌣ c) ⊗ (b ⌣ d)$ for cohomology classes a, c in H*(X) and b, d in H*(Y)

Advanced techniques and computations

• Spectral sequences can be used to compute cohomology rings of more complex spaces
  • Serre spectral sequence for fibrations relates the cohomology of the base, fiber, and total space
  • Eilenberg-Moore spectral sequence computes the cohomology of a pullback or pushout square
• Poincaré duality relates the cup product in cohomology to the intersection product in homology
  • For a closed, oriented n-manifold M, Poincaré duality states $Hᵏ(M) ≅ Hₙ₋ₖ(M)$
  • The cup product in cohomology corresponds to the intersection product in homology under this isomorphism
• Cohomology ring computations can be simplified using algebraic techniques
  • Graded ring isomorphisms (ℤ[α]/〈α²〉 ≅ ℤ[β] with |β| = 2|α|) can make computations more manageable
  • Graded ring invariants (Krull dimension, depth, regularity) provide information about the structure of the ring

Cohomology ring vs homology

Relationship between cohomology and homology

• The cohomology ring H*(X) is the graded ring obtained from the cohomology groups Hⁿ(X) with the cup product as multiplication
• The homology groups Hₙ(X) are the duals of the cohomology groups Hⁿ(X)
  • Hₙ(X) ≅ Hom(Hⁿ(X), ℤ), where Hom denotes the group of homomorphisms
  • Elements of Hₙ(X) can be viewed as functions that assign integers to cohomology classes in Hⁿ(X)
• The cup product in cohomology is dual to the intersection product in homology
  • For an n-dimensional manifold X, the intersection product is a bilinear map Hₚ(X) × Hᵩ(X) → Hₚ₊ᵩ₋ₙ(X)
  • The intersection product computes the homology class of the intersection of two submanifolds representing homology classes

Advantages of the cohomology ring

• The cohomology ring has additional structure due to its multiplicative properties
  • The cup product provides a way to multiply cohomology classes, while homology lacks a natural multiplication
  • The ring structure encodes higher-order relationships between cohomology classes that are not apparent from homology groups
• The cohomology ring is a finer invariant than homology groups
  • Spaces with isomorphic homology groups may have non-isomorphic cohomology rings (ℂP² and S² × S⁴)
  • The cohomology ring can detect subtle topological differences that homology alone cannot
• Cohomology rings are more naturally related to other algebraic structures
  • Cohomology rings are graded commutative, meaning $a ⌣ b = (-1)^{|a||b|} b ⌣ a$ for cohomology classes a and b
  • Cohomology rings are often isomorphic to well-known algebraic objects (polynomial rings, exterior algebras)

Distinguishing spaces with cohomology

Non-homotopy equivalent spaces

• If two spaces have non-isomorphic cohomology rings, then they cannot be homotopy equivalent
  • Homotopy equivalence implies isomorphic cohomology rings, so non-isomorphic rings prove spaces are not homotopy equivalent
  • Cohomology ring isomorphism is a necessary, but not sufficient, condition for homotopy equivalence
• The cohomology ring can detect differences in the multiplicative structure of cohomology classes
  • Spaces with isomorphic cohomology groups may have different cup product structures
  • Examples include ℂP² and S² × S⁴, which have isomorphic cohomology groups but non-isomorphic rings

Higher-order cohomology operations

• Massey products are higher-order cohomology operations that can distinguish spaces with isomorphic cohomology rings
  • Massey products are defined using the cup product and provide additional structure on the cohomology ring
  • Spaces with isomorphic cohomology rings may have different Massey product structures, indicating different homotopy types
• Steenrod squares are cohomology operations that provide additional information beyond the cup product
  • Steenrod squares are stable operations that map Hⁿ(X; ℤ/2ℤ) to Hⁿ⁺ⁱ(X; ℤ/2ℤ) for i ≥ 0
  • The action of Steenrod squares on the cohomology ring can distinguish spaces with isomorphic rings (real projective spaces)
• Higher-order cohomology operations are an active area of research in algebraic topology
  • New operations and structures are being discovered and applied to distinguish spaces and study their properties
  • The relationship between higher-order operations and other invariants (homotopy groups, K-theory) is an important topic of investigation