🍃Sheaf Theory Unit 5 – Čech cohomology

Key Concepts and Definitions

• Čech cohomology extends the idea of cohomology to more general spaces by using open covers
• Presheaves are contravariant functors from the category of open sets of a topological space to the category of abelian groups
• Sheaves are presheaves satisfying the gluing axiom and the locality axiom
  • Gluing axiom ensures that local data can be uniquely glued together to obtain global data
  • Locality axiom guarantees that a section of a sheaf over an open set is determined by its values on any open cover of that set
• Čech complex is a cochain complex constructed from an open cover of a topological space and a presheaf on that space
• Refinement of an open cover is another open cover such that each open set in the new cover is contained in some open set of the original cover
• Directed system is a collection of objects indexed by a directed set, along with morphisms between the objects that are compatible with the ordering of the indexing set

Historical Context and Development

• Čech cohomology was introduced by Eduard Čech in the 1930s as a way to extend cohomology theory to more general spaces
• Developed as a response to the limitations of singular cohomology, which works well for simplicial complexes but not for more general topological spaces
• Čech's original definition used infinite covers and inverse limits, which made it difficult to compute
• Later reformulated using finite open covers and direct limits, making it more accessible and computationally tractable
• Became an important tool in the development of sheaf theory and the study of global properties of topological spaces
• Čech cohomology played a crucial role in the proof of the de Rham theorem, which relates de Rham cohomology to singular cohomology

Čech Complexes and Coverings

• Given a topological space $X$ and an open cover $\mathcal{U} = \{U_i\}_{i \in I}$ of $X$, the Čech complex is a cochain complex $\check{C}^*(\mathcal{U}, \mathcal{F})$ associated to a presheaf $\mathcal{F}$ on $X$
• The $n$-th cochain group $\check{C}^n(\mathcal{U}, \mathcal{F})$ consists of alternating functions $\alpha$ that assign to each $(n+1)$-tuple $(i_0, \ldots, i_n)$ an element $\alpha(i_0, \ldots, i_n) \in \mathcal{F}(U_{i_0} \cap \ldots \cap U_{i_n})$
  • Alternating property means that the value changes sign whenever two indices are swapped
• The coboundary map $\delta^n: \check{C}^n(\mathcal{U}, \mathcal{F}) \to \check{C}^{n+1}(\mathcal{U}, \mathcal{F})$ is defined by the usual formula for the coboundary in a cochain complex
• Refining the open cover leads to a morphism between the corresponding Čech complexes, which induces a morphism on cohomology
• The Čech cohomology $\check{H}^*(\mathcal{U}, \mathcal{F})$ is defined as the cohomology of the Čech complex $\check{C}^*(\mathcal{U}, \mathcal{F})$
• Čech cohomology of $X$ with coefficients in $\mathcal{F}$ is obtained by taking the direct limit of $\check{H}^*(\mathcal{U}, \mathcal{F})$ over all open covers $\mathcal{U}$ of $X$

Computing Čech Cohomology

• To compute Čech cohomology, start by choosing an open cover $\mathcal{U} = \{U_i\}_{i \in I}$ of the topological space $X$
• Construct the Čech complex $\check{C}^*(\mathcal{U}, \mathcal{F})$ associated to the open cover and a presheaf $\mathcal{F}$ on $X$
  • The cochain groups are given by $\check{C}^n(\mathcal{U}, \mathcal{F}) = \prod_{i_0 < \ldots < i_n} \mathcal{F}(U_{i_0} \cap \ldots \cap U_{i_n})$
  • The coboundary maps are defined using the restriction maps of the presheaf
• Compute the cohomology of the Čech complex by finding the kernel and image of the coboundary maps
  • The $n$-th Čech cohomology group is given by $\check{H}^n(\mathcal{U}, \mathcal{F}) = \ker(\delta^n) / \operatorname{im}(\delta^{n-1})$
• Refine the open cover and repeat the process, obtaining a directed system of cohomology groups
• Take the direct limit of this directed system to obtain the Čech cohomology of $X$ with coefficients in $\mathcal{F}$
  • The direct limit captures the information that is preserved under refinement of open covers
• In practice, it is often sufficient to compute the Čech cohomology with respect to a specific open cover, especially if the cover is fine enough to capture the relevant topological features of the space

Relation to Sheaf Cohomology

• Čech cohomology is closely related to sheaf cohomology, which is another way of defining cohomology for sheaves on topological spaces
• For a sheaf $\mathcal{F}$ on a topological space $X$, the sheaf cohomology groups $H^*(X, \mathcal{F})$ are defined using injective resolutions of $\mathcal{F}$
• Čech cohomology provides a more concrete way to compute sheaf cohomology, using open covers and Čech complexes
• The Leray theorem states that for a paracompact Hausdorff space $X$ and a sheaf $\mathcal{F}$ on $X$, the Čech cohomology $\check{H}^*(X, \mathcal{F})$ is isomorphic to the sheaf cohomology $H^*(X, \mathcal{F})$
  • This isomorphism is natural with respect to morphisms of sheaves
• In many cases, Čech cohomology is easier to compute than sheaf cohomology, as it avoids the use of injective resolutions
• The relation between Čech and sheaf cohomology has been generalized to other settings, such as étale cohomology in algebraic geometry

Applications in Topology and Geometry

• Čech cohomology has numerous applications in topology and geometry, as it provides a way to study global properties of spaces using local data
• In algebraic topology, Čech cohomology is used to define cohomology theories for general topological spaces, extending the reach of singular cohomology
  • Čech cohomology is particularly useful for studying spaces that are not well-behaved, such as non-paracompact spaces or spaces with bad local properties
• In complex analysis and algebraic geometry, Čech cohomology is used to study the cohomology of sheaves on complex manifolds and algebraic varieties
  • The Dolbeault theorem relates Čech cohomology of holomorphic sheaves to Dolbeault cohomology, which is defined using differential forms
  • Čech cohomology is also used in the construction of the Picard group and the Brauer group of a scheme
• In differential geometry, Čech cohomology appears in the study of characteristic classes and obstruction theory
  • The Chern-Weil homomorphism relates Čech cohomology to the de Rham cohomology of a smooth manifold, allowing the construction of characteristic classes from connection forms
• Čech cohomology has also found applications in mathematical physics, particularly in the study of gauge theories and string theory
  • In these contexts, Čech cohomology is used to describe the global structure of gauge fields and to classify topological defects

Examples and Problem-Solving Techniques

• To compute the Čech cohomology of the circle $S^1$ with constant coefficients $\mathbb{Z}$, use the open cover $\mathcal{U} = \{U, V\}$ where $U$ and $V$ are slightly larger than semicircles
  • The Čech complex has non-zero terms $\check{C}^0(\mathcal{U}, \mathbb{Z}) = \mathbb{Z} \oplus \mathbb{Z}$ and $\check{C}^1(\mathcal{U}, \mathbb{Z}) = \mathbb{Z}$, with the coboundary map given by the difference of the restriction maps
  • The resulting cohomology groups are $\check{H}^0(S^1, \mathbb{Z}) = \mathbb{Z}$ and $\check{H}^1(S^1, \mathbb{Z}) = \mathbb{Z}$, agreeing with the singular cohomology of the circle
• For the Möbius band $M$, use the open cover $\mathcal{U} = \{U, V\}$ where $U$ and $V$ are slightly larger than the rectangular strips obtained by cutting the band along its centerline
  • The Čech complex with constant coefficients $\mathbb{Z}$ has terms $\check{C}^0(\mathcal{U}, \mathbb{Z}) = \mathbb{Z} \oplus \mathbb{Z}$ and $\check{C}^1(\mathcal{U}, \mathbb{Z}) = \mathbb{Z} \oplus \mathbb{Z}$, with the coboundary map given by the difference of the restriction maps, taking into account the twist in the Möbius band
  • The resulting cohomology groups are $\check{H}^0(M, \mathbb{Z}) = \mathbb{Z}$ and $\check{H}^1(M, \mathbb{Z}) = 0$, capturing the fact that the Möbius band is non-orientable
• When solving problems involving Čech cohomology, it is often helpful to choose an open cover that is well-suited to the space and the coefficients
  • For spaces with a simple local structure, such as manifolds or CW complexes, it is often sufficient to use a cover by open sets that are contractible or have a simple cohomology
  • For more complicated spaces, such as spaces with bad local properties or spaces with a non-trivial fundamental group, it may be necessary to use more sophisticated covers or to refine the cover adaptively
• It is also important to keep track of the relationships between different open covers and the induced maps on cohomology
  • Refining an open cover leads to a map between the corresponding Čech complexes, which induces a map on cohomology
  • Understanding these induced maps is crucial for computing the direct limit that defines the Čech cohomology of the space

Advanced Topics and Extensions

• Čech cohomology can be generalized to the setting of simplicial sheaves, where the open sets in the cover are replaced by simplicial objects
  • This generalization is useful in algebraic geometry, where it is used to define étale cohomology and other cohomology theories for schemes
• The Čech-to-derived functor spectral sequence relates Čech cohomology to derived functor cohomology, providing a way to compute sheaf cohomology using Čech cochains
  • This spectral sequence is a powerful tool for computing cohomology in various settings, including algebraic geometry and complex analysis
• Deligne cohomology is a generalization of Čech cohomology that incorporates differential forms and allows for the construction of secondary characteristic classes
  • Deligne cohomology has applications in the study of Chern-Simons theory and other gauge-theoretic problems
• Čech cohomology can also be extended to the setting of non-abelian cohomology, where the coefficients are taken in a non-abelian category, such as the category of groups or the category of sheaves of groups
  • Non-abelian Čech cohomology has applications in the study of principal bundles and gerbes, which are geometric objects that generalize line bundles and play a role in mathematical physics
• In the setting of algebraic topology, Čech cohomology is related to other cohomology theories, such as Alexander-Spanier cohomology and Steenrod-Sitnikov cohomology
  • These cohomology theories are defined using different types of covers and limits, and they provide alternative approaches to studying the global properties of topological spaces
• The relationship between Čech cohomology and other cohomology theories, such as de Rham cohomology and singular cohomology, has been the subject of extensive research
  • Understanding these relationships is important for computing cohomology in various settings and for understanding the connections between different branches of mathematics, such as topology, geometry, and analysis