2.1 Cohomology groups

Cohomology groups are powerful tools in algebraic topology, capturing essential information about a space's structure. They assign abelian groups to each dimension, revealing intrinsic topological features like "holes" or "voids" in the space.

These groups are defined using cochains, cocycles, and coboundaries. Properties like functoriality and long exact sequences make cohomology groups invaluable for studying spaces and their relationships. Various computational techniques and applications demonstrate their versatility in mathematics and physics.

Definition of cohomology groups

• Cohomology groups are algebraic objects associated to a topological space that capture essential information about its structure and properties
• They provide a way to study the "holes" or "voids" in a space by assigning abelian groups to each dimension, revealing intrinsic topological features

Cochains and coboundary operators

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• Cochains are dual objects to chains, assigning abelian groups to each dimension of a space
  • A $p$-cochain is a homomorphism from the group of $p$-chains to an abelian group $G$
• The coboundary operator $\delta$ maps $p$-cochains to $(p+1)$-cochains, satisfying $\delta \circ \delta = 0$
  • Analogous to the boundary operator for chains, but increasing dimension

Cocycles and coboundaries

• A $p$-cocycle is a $p$-cochain $\alpha$ such that $\delta \alpha = 0$
  • Represents a cohomology class that captures a "hole" in dimension $p$
• A $p$-coboundary is a $p$-cochain $\beta$ of the form $\beta = \delta \gamma$ for some $(p-1)$-cochain $\gamma$
  • Coboundaries are trivial cocycles, not carrying essential topological information

Cohomology groups as quotient groups

• The $p$-th cohomology group $H^p(X; G)$ is defined as the quotient group of $p$-cocycles modulo $p$-coboundaries
  • $H^p(X; G) = \ker(\delta_p) / \operatorname{im}(\delta_{p-1})$
• Elements of $H^p(X; G)$ are equivalence classes of cocycles, with two cocycles equivalent if they differ by a coboundary
  • Captures the essential "holes" in dimension $p$, modulo the trivial ones

Properties of cohomology groups

• Cohomology groups satisfy several important properties that make them powerful tools in algebraic topology
• These properties allow for the computation and comparison of cohomology groups in various settings

Functoriality of cohomology

• Cohomology is a contravariant functor from the category of topological spaces to the category of abelian groups
  • A continuous map $f: X \to Y$ induces a homomorphism $f^*: H^p(Y; G) \to H^p(X; G)$ for each $p$
• Functoriality allows for the study of maps between spaces via induced homomorphisms on cohomology

Long exact sequence in cohomology

• For a pair $(X, A)$ of a space $X$ and a subspace $A$, there is a long exact sequence relating the cohomology groups of $X$, $A$, and the relative cohomology $H^p(X, A; G)$
  • $\cdots \to H^{p-1}(A; G) \to H^p(X, A; G) \to H^p(X; G) \to H^p(A; G) \to \cdots$
• The long exact sequence is a powerful tool for computing cohomology groups and understanding the relationship between a space and its subspaces

Excision theorem and Mayer-Vietoris sequence

• The excision theorem states that the relative cohomology $H^p(X, A; G)$ is isomorphic to $H^p(X - U, A - U; G)$ for any open set $U \subset A$
  • Allows for the computation of relative cohomology by "excising" a suitable subset
• The Mayer-Vietoris sequence is a long exact sequence relating the cohomology of a space $X$ to the cohomology of two open subsets $U, V$ covering $X$
  • Provides a method for computing cohomology by breaking a space into simpler pieces

Cohomology with coefficients

• Cohomology groups can be defined with various coefficient groups, leading to different flavors of cohomology with additional structure and properties
• The choice of coefficients can provide more refined information about the topology of a space

Cohomology with constant coefficients

• The most basic form of cohomology, where the coefficient group $G$ is a fixed abelian group
  • Captures the global topological features of a space
• Cohomology with constant coefficients satisfies all the standard properties, such as functoriality and long exact sequences

Cohomology with local coefficients

• A generalization of cohomology where the coefficient group varies over the space, forming a local system
  • Allows for the study of spaces with non-trivial fundamental group, such as non-orientable manifolds
• Local coefficients can encode additional topological and geometric information, such as orientation or twisting

Universal coefficient theorem

• A theorem relating cohomology with different coefficient groups, stating that there is a short exact sequence
  • $0 \to \operatorname{Ext}(H_{p-1}(X; \mathbb{Z}), G) \to H^p(X; G) \to \operatorname{Hom}(H_p(X; \mathbb{Z}), G) \to 0$
• The theorem allows for the computation of cohomology with arbitrary coefficients from integral homology and provides a classification of cohomology groups

Cup product in cohomology

• The cup product is an additional structure on cohomology groups, providing a multiplicative operation that is compatible with the additive structure
• It turns the direct sum of cohomology groups into a graded ring, revealing deeper topological and algebraic properties

Definition and properties of cup product

• The cup product of two cochains $\alpha \in C^p(X; G)$ and $\beta \in C^q(X; H)$ is a cochain $\alpha \smile \beta \in C^{p+q}(X; G \otimes H)$
  • Defined using the diagonal map and the tensor product of coefficient groups
• The cup product is associative, distributive over addition, and compatible with the coboundary operator
  • Induces a well-defined product on cohomology groups

Cohomology rings and graded-commutativity

• The cup product turns the direct sum of cohomology groups $H^*(X; R) = \bigoplus_p H^p(X; R)$ into a graded ring
  • The grading is given by the dimension of the cohomology groups
• The cohomology ring is graded-commutative, satisfying $\alpha \smile \beta = (-1)^{pq} \beta \smile \alpha$ for $\alpha \in H^p(X; R)$ and $\beta \in H^q(X; R)$
  • Reflects the underlying commutativity of the cup product at the cochain level

Künneth formula for cohomology

• A theorem describing the cohomology of a product space $X \times Y$ in terms of the cohomology of $X$ and $Y$
  • States that there is an isomorphism of graded rings $H^*(X \times Y; R) \cong H^*(X; R) \otimes H^*(Y; R)$
• The Künneth formula allows for the computation of cohomology rings of product spaces and provides insight into the multiplicative structure of cohomology

Poincaré duality and cohomology

• Poincaré duality is a fundamental theorem relating cohomology and homology of orientable manifolds
• It provides a deep connection between the algebraic and geometric properties of a manifold

Orientation and fundamental class

• An orientation of an $n$-dimensional manifold $M$ is a consistent choice of generator for the top homology group $H_n(M; \mathbb{Z})$
  • Corresponds to a choice of "positive" direction or volume form on the manifold
• The fundamental class $[M] \in H_n(M; \mathbb{Z})$ is the chosen generator representing the orientation
  • Serves as a canonical element for Poincaré duality

Statement and proof of Poincaré duality

• Poincaré duality states that 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 isomorphism is given by the cap product with the fundamental class $[M]$
• The proof of Poincaré duality involves the construction of a dual cell decomposition and the use of the cap product and the Kronecker pairing
  • Relies on the orientability of the manifold and the properties of the fundamental class

Poincaré duality for non-compact manifolds

• Poincaré duality can be extended to non-compact orientable manifolds with suitable modifications
  • Requires the use of cohomology with compact support and homology with closed support
• For a non-compact, orientable $n$-manifold $M$, there is an isomorphism $H^k_c(M; R) \cong H_{n-k}(M; R)$
  • Relates cohomology with compact support and ordinary homology
• Poincaré duality for non-compact manifolds allows for the study of the cohomology of open manifolds and manifolds with boundary

Computational techniques for cohomology

• Various computational techniques have been developed to calculate cohomology groups in different settings
• These techniques often rely on additional structures or properties of the spaces involved

Cellular cohomology and CW complexes

• Cellular cohomology is a method for computing cohomology groups of CW complexes using the cellular chain complex
  • A CW complex is a space built by attaching cells of increasing dimension via attaching maps
• The cellular cochain complex is dual to the cellular chain complex, with coboundary maps induced by the attaching maps
  • Cellular cohomology groups are the cohomology groups of this cochain complex
• Cellular cohomology provides a combinatorial approach to computing cohomology, reducing it to linear algebra over the coefficient group

de Rham cohomology and differential forms

• de Rham cohomology is a cohomology theory for smooth manifolds based on differential forms
  • A differential $k$-form is a smooth section of the $k$-th exterior power of the cotangent bundle
• The de Rham complex is the cochain complex of differential forms with the exterior derivative as the coboundary operator
  • de Rham cohomology groups are the cohomology groups of this complex
• de Rham's theorem states that de Rham cohomology is isomorphic to singular cohomology with real coefficients
  • Provides a link between the algebraic and analytic aspects of cohomology

Čech cohomology and sheaf cohomology

• Čech cohomology is a cohomology theory based on open covers of a space and their intersections
  • Defined using Čech cochains, which assign values to intersections of open sets in a cover
• The Čech cochain complex is constructed using the coboundary maps induced by the inclusion of intersections
  • Čech cohomology groups are the cohomology groups of this complex
• Sheaf cohomology is a generalization of Čech cohomology that uses sheaves instead of open covers
  • A sheaf is a data structure assigning abelian groups to open sets, with compatibility conditions
• Sheaf cohomology provides a more abstract and versatile framework for studying cohomology, with applications in algebraic geometry and complex analysis

Applications of cohomology groups

• Cohomology groups have numerous applications in various branches of mathematics and theoretical physics
• They provide powerful tools for studying geometric, topological, and algebraic structures

Characteristic classes and vector bundles

• Characteristic classes are cohomology classes associated to vector bundles, measuring their non-triviality
  • Examples include Chern classes, Pontryagin classes, and Euler classes
• Characteristic classes provide obstructions to the existence of certain structures on vector bundles (orientations, spin structures, almost complex structures)
  • Computed using the classifying space and the universal bundle construction
• Characteristic classes have applications in differential geometry, algebraic topology, and mathematical physics (gauge theory, string theory)

Obstruction theory and extension problems

• Obstruction theory studies the existence and classification of extensions of continuous functions or cross-sections of bundles
  • Uses cohomology classes as obstructions to the existence of such extensions
• The obstruction to extending a continuous function $f: A \to Y$ to $X \supset A$ lies in the cohomology group $H^{n+1}(X, A; \pi_n(Y))$
  • Vanishing of the obstruction class implies the existence of an extension
• Obstruction theory has applications in homotopy theory, fiber bundle theory, and the classification of manifolds

Cohomological dimension and embedding theorems

• The cohomological dimension of a space $X$ is the smallest integer $n$ such that $H^k(X; G) = 0$ for all $k > n$ and all coefficient groups $G$
  • Measures the complexity of the space from a cohomological perspective
• Embedding theorems relate the cohomological dimension of a space to its embeddability into Euclidean spaces
  • Examples include the Whitney embedding theorem and the Haefliger-Hirsch theorem
• Cohomological dimension has applications in manifold theory, group theory, and the study of geometric structures on spaces