🔢Algebraic Topology Unit 7 – Cohomology and Cup Product

Key Concepts and Definitions

• Cohomology is a dual theory to homology, assigning algebraic objects (abelian groups) to a topological space
• Cochains are homomorphisms from chain groups to a coefficient group, typically a field or ring
• The coboundary operator $\delta$ raises the degree of a cochain by 1 and satisfies $\delta^2 = 0$
  • Analogous to the boundary operator in homology
• Cocycles are cochains $\alpha$ such that $\delta \alpha = 0$, forming the kernel of $\delta$
• Coboundaries are cochains $\beta$ such that $\beta = \delta \gamma$ for some cochain $\gamma$, forming the image of $\delta$
• The $n$-th cohomology group $H^n(X; G)$ is defined as the quotient of $n$-cocycles by $n$-coboundaries
  • Measures the "holes" in the space $X$ that $n$-dimensional cochains can detect
• The cup product is a bilinear operation that combines two cochains to produce a higher-degree cochain

Historical Context and Motivation

• Cohomology was developed in the 1930s and 1940s by mathematicians such as J.W. Alexander, A. Kolmogorov, and N. Čech
• Emerged as a way to study topological spaces by assigning algebraic invariants
• Motivation came from the desire to understand duality in topology and to find a cohomological analogue of the intersection product in homology
• The cup product, introduced by E. Čech and H. Whitney, provided a means to multiply cohomology classes
• Cohomology and the cup product have found applications in various areas of mathematics, including:
  • Algebraic geometry (sheaf cohomology)
  • Differential geometry (de Rham cohomology)
  • Mathematical physics (gauge theory, string theory)
• They have also been used to study and classify topological spaces, such as manifolds and CW complexes

Cochain Complexes and Cohomology Groups

• A cochain complex is a sequence of abelian groups $C^n$ (cochain groups) connected by homomorphisms $\delta^n: C^n \to C^{n+1}$ (coboundary operators) such that $\delta^{n+1} \circ \delta^n = 0$
• The cochain groups $C^n$ are typically defined using homomorphisms from the chain groups $C_n$ to a coefficient group $G$
• The coboundary operator $\delta$ is induced by the boundary operator $\partial$ in the chain complex
  • For a cochain $\alpha \in C^n$ and a chain $c \in C_{n+1}$, $(\delta \alpha)(c) = \alpha(\partial c)$
• The cohomology groups $H^n(C^*) = \ker(\delta^n) / \operatorname{im}(\delta^{n-1})$ measure the "obstruction" to extending cochains
• Cohomology groups are contravariant functors, meaning they reverse the direction of maps between spaces
• The universal coefficient theorem relates homology and cohomology groups via a short exact sequence involving the Ext functor

Properties of Cohomology

• Cohomology is a contravariant functor from the category of topological spaces to the category of abelian groups
• Homotopy invariance: If two spaces $X$ and $Y$ are homotopy equivalent, then their cohomology groups are isomorphic
• Excision: For a "good" pair $(X, A)$, there is an isomorphism between the relative cohomology $H^n(X, A; G)$ and $H^n(X / A, *; G)$
• Long exact sequence: A short exact sequence of cochain complexes induces a long exact sequence in cohomology
• Künneth formula: For spaces $X$ and $Y$, there is a split short exact sequence involving the tensor product of their cohomology groups
• Cohomology with compact supports: A variant of cohomology that captures the behavior "at infinity" for non-compact spaces
• Poincaré duality: For orientable $n$-manifolds, there is an isomorphism between $H^k(M; G)$ and $H_{n-k}(M; G)$

The Cup Product: Definition and Intuition

• 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
• Intuition: The cup product measures the "twisting" or "linking" of cocycles
  • If two cocycles $\alpha$ and $\beta$ can be "unlinked," their cup product is zero
• Definition: For cochains $\alpha \in C^p(X; R)$ and $\beta \in C^q(X; R)$, the cup product $\alpha \smile \beta \in C^{p+q}(X; R)$ is given by:
  • $(\alpha \smile \beta)(\sigma) = \alpha(\sigma_{[0, p]}) \cdot \beta(\sigma_{[p, p+q]})$ for a simplex $\sigma: \Delta^{p+q} \to X$
• The cup product is well-defined on cohomology classes and is independent of the choice of representative cocycles
• Properties:
  • Associative: $(\alpha \smile \beta) \smile \gamma = \alpha \smile (\beta \smile \gamma)$
  • Graded commutative: $\alpha \smile \beta = (-1)^{pq} \beta \smile \alpha$
  • Functorial: For a map $f: X \to Y$, $f^*(\alpha \smile \beta) = f^*(\alpha) \smile f^*(\beta)$
• The cup product provides a ring structure on the direct sum of cohomology groups $H^*(X; R) = \bigoplus_n H^n(X; R)$

Computing Cup Products

• To compute the cup product of two cohomology classes, choose representative cocycles and apply the definition
• For simplicial cohomology, the cup product can be computed using the simplicial cochain complex
  • Evaluate the cocycles on the front and back faces of simplices and multiply the results
• For singular cohomology, the cup product is computed using the singular cochain complex
  • Subdivide simplices and evaluate the cocycles on the resulting pieces
• The Alexander-Whitney map provides a chain homotopy equivalence between the simplicial and singular cochain complexes, allowing for consistent computations
• For CW complexes, the cellular cochain complex can be used to compute cup products
  • The cup product of cellular cochains is determined by the incidence relations between cells
• In practice, it is often easier to compute cup products using known properties and relations, such as:
  • The cup product of a cocycle with a coboundary is zero
  • The cup product of a generator with itself is determined by the cohomology ring structure
• Poincaré duality can be used to relate cup products in complementary dimensions for manifolds

Applications in Topology and Beyond

• The cup product is a powerful tool for studying the algebraic topology of spaces
• Cohomology rings: The cup product gives the direct sum of cohomology groups $H^*(X; R)$ the structure of a graded-commutative ring
  • The cohomology ring encodes important topological information about the space $X$
• Characteristic classes: The cup product is used to define and compute characteristic classes of vector bundles, such as:
  • Stiefel-Whitney classes in mod 2 cohomology
  • Chern classes in integral cohomology
  • Pontryagin classes in rational cohomology
• Obstruction theory: The cup product appears in the obstruction cocycle for extending maps and homotopies
• Massey products: Higher-order cohomology operations that generalize the cup product and provide finer topological invariants
• In physics, the cup product is related to the wedge product of differential forms and is used in the formulation of various theories, such as:
  • Chern-Simons theory
  • Wess-Zumino-Witten model
  • Topological quantum field theories
• Cup products also appear in the study of group cohomology, Lie algebra cohomology, and Hochschild cohomology

Common Challenges and Problem-Solving Strategies

• Computing cup products can be challenging, especially for large cochain complexes or complicated spaces
  • Break the problem into smaller pieces by using the properties of the cup product and the structure of the space
  • Look for ways to simplify the cochain complex, such as using a CW structure or collapsing contractible subcomplexes
• Understanding the geometric meaning of the cup product can be difficult
  • Think about how cocycles "twist" or "link" together
  • Consider the cup product in low dimensions and for simple spaces, such as spheres and tori
• Determining the cohomology ring structure can be a complex task
  • Use known results for common spaces, such as projective spaces and Grassmannians
  • Apply the Künneth formula to compute the cohomology of product spaces
  • Utilize Poincaré duality to relate cup products in complementary dimensions for manifolds
• Working with non-trivial coefficients can add complexity to computations
  • Be mindful of the action of the fundamental group on the coefficients
  • Use the universal coefficient theorem to relate cohomology with different coefficients
• Dealing with torsion in cohomology can be tricky
  • Consider using field coefficients to simplify computations
  • Keep track of the torsion using the structure of the cohomology groups
• When stuck, try to find a similar problem or example that has been solved before
  • Look for analogies with other cohomology theories, such as de Rham cohomology or sheaf cohomology
  • Consult textbooks, research papers, and online resources for guidance and inspiration