4.4 Alexandrov-Čech cohomology

Alexandrov-Čech cohomology assigns abelian groups to topological spaces, measuring "holes" and "obstructions." It approximates spaces using open covers, providing insights into their structure and properties. This theory is closely related to singular and sheaf cohomology.

The construction involves nerves of open covers, Čech complexes, and cohomology groups. Key properties include homotopy invariance, excision theorem, and long exact sequences. Alexandrov-Čech cohomology is useful for computing cohomology groups of spheres, projective spaces, and manifolds.

Definition of Alexandrov-Čech cohomology

• Alexandrov-Čech cohomology is a cohomology theory in algebraic topology that assigns abelian groups to topological spaces
• It is based on the idea of approximating a space by its open covers and studying the relationships between these covers
• Alexandrov-Čech cohomology provides a way to measure the "holes" or "obstructions" in a topological space

Relation to singular cohomology

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• Alexandrov-Čech cohomology is closely related to singular cohomology, another important cohomology theory in algebraic topology
• For "nice" spaces (such as CW complexes), Alexandrov-Čech cohomology and singular cohomology give isomorphic cohomology groups
• However, Alexandrov-Čech cohomology can be defined for more general spaces where singular cohomology may not be well-behaved

Comparison with sheaf cohomology

• Alexandrov-Čech cohomology can also be compared with sheaf cohomology, a cohomology theory that assigns abelian groups to sheaves on a topological space
• In certain cases, such as for paracompact Hausdorff spaces, Alexandrov-Čech cohomology coincides with the cohomology of the constant sheaf
• The relationship between Alexandrov-Čech cohomology and sheaf cohomology is important in the study of cohomology theories and their applications

Construction of Alexandrov-Čech cohomology

• The construction of Alexandrov-Čech cohomology involves several key steps and objects
• It begins with the notion of an open cover of a topological space, which is a collection of open sets whose union is the entire space

Nerves of open covers

• Given an open cover $\mathcal{U}$ of a topological space $X$, the nerve of $\mathcal{U}$ is a simplicial complex $N(\mathcal{U})$
• The vertices of $N(\mathcal{U})$ correspond to the open sets in $\mathcal{U}$, and simplices correspond to finite intersections of these open sets
• The nerve construction provides a way to associate a simplicial complex to an open cover

Čech complex

• The Čech complex $\check{C}^*(\mathcal{U}, \mathcal{F})$ is a cochain complex associated to an open cover $\mathcal{U}$ and a sheaf $\mathcal{F}$ on a topological space $X$
• It is defined using the nerve of the open cover and the sections of the sheaf over the open sets in the cover
• The coboundary maps in the Čech complex are induced by the restriction maps of the sheaf

Cohomology groups

• The cohomology groups of the Čech complex, denoted by $\check{H}^*(\mathcal{U}, \mathcal{F})$, are the cohomology groups of the cochain complex $\check{C}^*(\mathcal{U}, \mathcal{F})$
• These groups measure the global sections of the sheaf $\mathcal{F}$ that are compatible with the open cover $\mathcal{U}$
• The Alexandrov-Čech cohomology groups of a topological space $X$ with coefficients in a sheaf $\mathcal{F}$ are defined as the direct limit of the cohomology groups $\check{H}^*(\mathcal{U}, \mathcal{F})$ over all open covers $\mathcal{U}$ of $X$

Properties of Alexandrov-Čech cohomology

• Alexandrov-Čech cohomology satisfies several important properties that make it a powerful tool in algebraic topology
• These properties allow for the computation and comparison of cohomology groups in various settings

Homotopy invariance

• Alexandrov-Čech cohomology is homotopy invariant, meaning that homotopy equivalent spaces have isomorphic Alexandrov-Čech cohomology groups
• This property allows for the study of topological spaces up to homotopy equivalence, which is a weaker notion than homeomorphism
• Homotopy invariance is a crucial property that simplifies many computations and arguments in algebraic topology

Excision theorem

• The excision theorem for Alexandrov-Čech cohomology states that the cohomology of a space can be computed locally
• More precisely, if $A$ is a closed subspace of a topological space $X$ and $U$ is an open set containing $A$, then the Alexandrov-Čech cohomology of the pair $(X, A)$ is isomorphic to the Alexandrov-Čech cohomology of the pair $(U, A)$
• The excision theorem is a powerful tool for breaking down the computation of cohomology groups into simpler pieces

Long exact sequence

• Alexandrov-Čech cohomology admits a long exact sequence for pairs of topological spaces
• Given a pair $(X, A)$ where $A$ is a closed subspace of $X$, there is a long exact sequence relating the Alexandrov-Čech cohomology groups of $X$, $A$, and the relative cohomology groups of the pair $(X, A)$
• This long exact sequence is a fundamental tool in algebraic topology for computing and relating cohomology groups

Functoriality

• Alexandrov-Čech cohomology is functorial, meaning that continuous maps between topological spaces induce homomorphisms between their cohomology groups
• This functoriality is compatible with the composition of continuous maps, making Alexandrov-Čech cohomology a contravariant functor from the category of topological spaces to the category of graded abelian groups
• Functoriality allows for the study of topological spaces and their relationships using the language of category theory

Computations using Alexandrov-Čech cohomology

• Alexandrov-Čech cohomology can be used to compute the cohomology groups of various important topological spaces
• These computations often rely on the properties and tools developed in the study of Alexandrov-Čech cohomology

Cohomology of spheres

• The Alexandrov-Čech cohomology groups of the $n$-dimensional sphere $S^n$ can be computed using the nerve of a suitable open cover
• For $n \geq 1$, the cohomology groups are given by $\check{H}^k(S^n) \cong \mathbb{Z}$ for $k = 0, n$ and $\check{H}^k(S^n) \cong 0$ for $0 < k < n$
• These computations demonstrate the relationship between the cohomology groups and the "holes" in the sphere

Cohomology of projective spaces

• Alexandrov-Čech cohomology can also be used to compute the cohomology groups of real and complex projective spaces
• For the real projective space $\mathbb{R}P^n$, the cohomology groups are given by $\check{H}^k(\mathbb{R}P^n) \cong \mathbb{Z}/2\mathbb{Z}$ for $0 \leq k \leq n$ and $\check{H}^k(\mathbb{R}P^n) \cong 0$ for $k > n$
• For the complex projective space $\mathbb{C}P^n$, the cohomology groups are given by $\check{H}^{2k}(\mathbb{C}P^n) \cong \mathbb{Z}$ for $0 \leq k \leq n$ and $\check{H}^{odd}(\mathbb{C}P^n) \cong 0$

Cohomology of manifolds

• Alexandrov-Čech cohomology is particularly useful for studying the cohomology of manifolds
• For a compact, connected, orientable manifold $M$ of dimension $n$, the top cohomology group $\check{H}^n(M)$ is isomorphic to $\mathbb{Z}$, and the other cohomology groups provide information about the topology of the manifold
• Poincaré duality, which relates the cohomology groups of a manifold to its homology groups, can be formulated using Alexandrov-Čech cohomology

Applications of Alexandrov-Čech cohomology

• Alexandrov-Čech cohomology has numerous applications in algebraic topology and related fields
• These applications demonstrate the power and versatility of cohomology as a tool for studying topological spaces and their properties

Obstruction theory

• Obstruction theory is a branch of algebraic topology that studies the existence and properties of continuous maps between topological spaces
• Alexandrov-Čech cohomology plays a central role in obstruction theory, as the obstructions to extending continuous maps are often formulated in terms of cohomology classes
• The vanishing of certain cohomology groups can provide conditions for the existence of continuous maps with desired properties

Characteristic classes

• Characteristic classes are cohomology classes associated to vector bundles and principal bundles over topological spaces
• Alexandrov-Čech cohomology can be used to define and study characteristic classes, such as Chern classes for complex vector bundles and Stiefel-Whitney classes for real vector bundles
• Characteristic classes provide important invariants of vector bundles and are related to the topology of the base space

Classifying spaces

• Classifying spaces are topological spaces that classify principal bundles with a given structure group
• The cohomology of classifying spaces, often computed using Alexandrov-Čech cohomology, is closely related to the characteristic classes of the associated bundles
• The study of classifying spaces and their cohomology is a central topic in algebraic topology and has applications in various areas of mathematics and physics

Spectral sequences

• Spectral sequences are algebraic tools that allow for the computation of cohomology groups by successively approximating them with simpler groups
• The Čech-to-derived functor spectral sequence, which relates Alexandrov-Čech cohomology to sheaf cohomology, is an important example of a spectral sequence in algebraic topology
• Spectral sequences provide a powerful framework for computing cohomology groups and understanding their relationships

Relation to other cohomology theories

• Alexandrov-Čech cohomology is one of several cohomology theories in algebraic topology, each with its own strengths and applications
• Understanding the relationships between these cohomology theories is an important aspect of the study of algebraic topology

Comparison with de Rham cohomology

• De Rham cohomology is a cohomology theory defined for smooth manifolds using differential forms
• For a smooth manifold $M$, the de Rham theorem states that the de Rham cohomology groups are isomorphic to the Alexandrov-Čech cohomology groups with real coefficients
• This relationship between Alexandrov-Čech cohomology and de Rham cohomology highlights the connection between topology and differential geometry

Comparison with étale cohomology

• Étale cohomology is a cohomology theory in algebraic geometry that is defined using the étale topology on schemes
• There is a comparison theorem relating étale cohomology to Alexandrov-Čech cohomology for complex algebraic varieties
• This comparison allows for the transfer of results between algebraic geometry and algebraic topology, and highlights the deep connections between these fields

Alexandrov-Čech cohomology vs singular cohomology

• Singular cohomology is another important cohomology theory in algebraic topology, defined using cochains on the singular simplices of a topological space
• For CW complexes, Alexandrov-Čech cohomology and singular cohomology give isomorphic cohomology groups
• However, Alexandrov-Čech cohomology can be defined for more general spaces and has certain advantages in terms of its properties and computations

Limitations and extensions

• While Alexandrov-Čech cohomology is a powerful tool in algebraic topology, it also has certain limitations and has been extended and generalized in various ways

Limitations of Alexandrov-Čech cohomology

• One limitation of Alexandrov-Čech cohomology is that it may not be well-suited for studying spaces with poor local properties, such as non-paracompact spaces
• Additionally, the computation of Alexandrov-Čech cohomology groups can be difficult in practice, as it involves taking a direct limit over all open covers of a space

Generalizations and extensions

• Several generalizations and extensions of Alexandrov-Čech cohomology have been developed to address its limitations and expand its scope
• These include Čech cohomology with compact supports, which is better suited for studying non-compact spaces, and Čech-Alexander-Spanier cohomology, which combines aspects of Alexandrov-Čech cohomology and Alexander-Spanier cohomology
• Other extensions, such as equivariant Čech cohomology and Čech cohomology with coefficients in a sheaf, provide additional tools for studying topological spaces with extra structure

Alternative approaches to cohomology

• In addition to the generalizations and extensions of Alexandrov-Čech cohomology, there are also alternative approaches to cohomology that provide different perspectives and tools
• These include sheaf cohomology, which is based on the theory of sheaves and provides a more general framework for studying cohomology theories
• Other approaches, such as simplicial cohomology and cellular cohomology, offer alternative methods for computing cohomology groups and understanding their properties