🧬Homological Algebra Unit 9 – Cohomology Theories

Introduction to Cohomology

• Cohomology is a powerful tool in algebraic topology that assigns algebraic invariants (abelian groups or rings) to topological spaces
• Dual concept to homology, capturing global properties of spaces by studying cocycles and coboundaries
• Provides a systematic way to study and classify topological spaces up to homotopy equivalence
• Cohomology theories have applications beyond topology, including in algebraic geometry, number theory, and physics
• Developed in the early 20th century by mathematicians such as J.W. Alexander, S. Eilenberg, and N. Steenrod
• Generalizes the notion of differential forms and de Rham cohomology to arbitrary topological spaces
• Allows for the computation of obstruction classes and characteristic classes of vector bundles

Fundamental Concepts and Definitions

• Cochain complexes are sequences of abelian groups or modules connected by coboundary maps satisfying $d^{n+1} \circ d^n = 0$
  • Cochains are dual to chains in homology, assigning algebraic objects to simplices or cells
  • Coboundary maps raise the degree of cochains and satisfy the coboundary condition
• Cocycles are cochains $c$ satisfying $d(c) = 0$, generalizing the notion of closed differential forms
  • Cocycles represent cohomology classes and capture global properties of the space
• Coboundaries are cochains of the form $d(b)$ for some cochain $b$, generalizing exact differential forms
  • Coboundaries are trivial cocycles and represent the zero cohomology class
• Cohomology groups are quotients of the group of cocycles by the group of coboundaries, $H^n(X; G) = Z^n(X; G) / B^n(X; G)$
  • Measure the failure of the coboundary condition and capture higher-order global information
• Cup product is a multiplicative structure on cohomology, turning it into a graded ring
  • Defined using the Alexander-Whitney map and the tensor product of cochains
• Cohomological operations, such as Steenrod squares and power operations, provide additional structure on cohomology

Types of Cohomology Theories

• Singular cohomology is defined using singular cochains on a topological space
  • Dual to singular homology and computable using simplicial approximation
• Čech cohomology is defined using open covers and the Čech nerve construction
  • Computes sheaf cohomology and is related to the Čech-de Rham complex
• de Rham cohomology is defined for smooth manifolds using differential forms and the exterior derivative
  • Isomorphic to singular cohomology via the de Rham theorem
• Sheaf cohomology is defined for sheaves on a topological space, capturing local-to-global properties
  • Generalizes Čech cohomology and is related to the derived functors of the global sections functor
• Étale cohomology is defined for schemes in algebraic geometry using étale topology and sheaves
  • Provides a cohomological approach to studying algebraic varieties and their arithmetic properties
• Group cohomology is defined for groups using cochains on the classifying space or the bar resolution
  • Captures information about group extensions, crossed homomorphisms, and group actions
• Galois cohomology is a special case of group cohomology for Galois groups of field extensions
  • Relates to the Brauer group, the Tate-Shafarevich group, and the Weil conjectures

Cohomology Groups and Their Properties

• Cohomology groups are contravariant functors from the category of topological spaces to the category of abelian groups
  • Induced homomorphisms $f^*: H^n(Y; G) \to H^n(X; G)$ for continuous maps $f: X \to Y$
  • Contravariance reflects the reversal of direction when pulling back cochains
• Long exact sequence of a pair relates the cohomology of a space, a subspace, and the quotient space
  • Generalizes the Mayer-Vietoris sequence and allows for computations using excision
• Künneth formula expresses the cohomology of a product space in terms of the cohomology of its factors and their tensor products
  • Involves the torsion product and the universal coefficient theorem for cohomology
• Poincaré duality relates the cohomology of a compact oriented manifold to its homology in complementary degrees
  • Realized by the cap product with the fundamental class and the evaluation map
• Cohomology with compact supports is a variant that captures the cohomology "at infinity" for non-compact spaces
  • Satisfies Poincaré duality for non-compact manifolds and is related to Borel-Moore homology
• Relative cohomology is defined for pairs of spaces and fits into long exact sequences
  • Allows for the definition of the connecting homomorphism and the study of relative invariants

Computational Techniques

• Mayer-Vietoris sequence is a long exact sequence relating the cohomology of a space to the cohomology of its subspaces
  • Allows for computations by decomposing a space into simpler pieces (open sets or CW complexes)
• Spectral sequences are algebraic tools for computing cohomology groups by successive approximations
  • Examples include the Leray spectral sequence, the Serre spectral sequence, and the Atiyah-Hirzebruch spectral sequence
• Cohomological spectral sequences often arise from filtrations or double complexes
  • Convergence properties depend on the structure of the spectral sequence (first or second quadrant, bounded or unbounded)
• Comparison theorems relate different cohomology theories under suitable hypotheses
  • Examples include the de Rham theorem, the Čech-de Rham isomorphism, and the comparison between étale and singular cohomology
• Characteristic classes, such as Chern classes, Stiefel-Whitney classes, and Pontryagin classes, are computable invariants of vector bundles
  • Computed using the Chern-Weil theory, the splitting principle, or the Leray-Hirsch theorem
• Cohomology operations, such as Steenrod squares and Massey products, provide additional computational tools
  • Used to detect non-trivial cohomology classes and to study the structure of the cohomology ring

Applications in Mathematics

• Algebraic topology: Cohomology is a fundamental tool for classifying topological spaces and studying their invariants
  • Used to define and compute characteristic classes, obstruction classes, and homotopy groups
• Algebraic geometry: Cohomology theories, such as sheaf cohomology and étale cohomology, are essential in the study of algebraic varieties
  • Used to prove the Weil conjectures, study the geometry of moduli spaces, and classify vector bundles
• Number theory: Cohomological methods, such as Galois cohomology and étale cohomology, have applications in arithmetic geometry
  • Used to study the Brauer group, the Tate-Shafarevich group, and the arithmetic of elliptic curves
• Mathematical physics: Cohomology appears in various contexts, such as gauge theory, string theory, and quantum field theory
  • Used to describe the topology of configuration spaces, classify instantons, and study anomalies
• Differential geometry: de Rham cohomology and Dolbeault cohomology are important tools in the study of smooth manifolds and complex manifolds
  • Used to study the geometry of vector bundles, characteristic classes, and the Atiyah-Singer index theorem
• Representation theory: Group cohomology and Lie algebra cohomology have applications in the study of representations
  • Used to classify group extensions, describe the structure of Lie algebras, and study the cohomology of homogeneous spaces

Connections to Other Algebraic Structures

• Cohomology theories often have a ring structure given by the cup product, turning them into graded commutative algebras
  • The cohomology ring encodes multiplicative information and is related to the Yoneda product in derived categories
• Massey products provide a higher-order generalization of the cup product, capturing additional multiplicative structure
  • Defined using cochains and the coboundary operator, and related to the Toda bracket in homotopy theory
• Cohomology operations, such as Steenrod squares and power operations, are cohomology classes in the cohomology of the Eilenberg-MacLane spaces
  • Give rise to additional algebraic structures, such as Steenrod algebras and unstable algebras over them
• Cohomology theories can be generalized to extraordinary cohomology theories, such as K-theory and cobordism theory
  • These theories have a more intricate structure, including products, operations, and Adams spectral sequences
• Sheaf cohomology is related to the derived functors of the global sections functor and fits into the framework of homological algebra
  • Connects cohomology to the theory of derived categories, derived functors, and spectral sequences
• Hochschild cohomology and cyclic cohomology are cohomology theories for algebras and provide invariants in noncommutative geometry
  • Related to the deformation theory of algebras and the study of algebraic K-theory

Advanced Topics and Current Research

• Motivic cohomology is a cohomology theory for algebraic varieties that combines algebraic and topological information
  • Defined using algebraic cycles and related to the study of special values of L-functions and the Beilinson conjectures
• Equivariant cohomology is a cohomology theory for spaces with group actions, taking into account the symmetries of the space
  • Defined using equivariant sheaves or the Borel construction, and related to the study of group actions and quotient spaces
• Quantum cohomology is a deformation of the ordinary cohomology ring of a symplectic manifold, incorporating information from pseudoholomorphic curves
  • Plays a central role in mirror symmetry and the study of Gromov-Witten invariants
• Nonabelian cohomology is a generalization of cohomology that allows for coefficients in nonabelian groups or groupoids
  • Arises in the study of gerbes, principal bundles, and higher gauge theory
• Homotopy coherent cohomology theories are a framework for studying cohomology theories that are compatible with homotopy theory
  • Defined using $\infty$-categories and related to the study of spectra and stable homotopy theory
• Persistent cohomology is a tool from applied algebraic topology that studies the evolution of cohomology classes across a filtration
  • Used in topological data analysis to study the shape and structure of data sets
• Categorification of cohomology theories aims to lift cohomology groups to higher categorical structures, such as chain complexes or spectra
  • Provides a deeper understanding of the structure of cohomology and its relationship to other invariants